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.

176

The definite and the indefinite integral.

Bk. III. ch. I.

Thus as a particular case we can conceive the interval from a to
x broken up into equal parts Ax, of which the number n increases
without limit; the required value is then expressed in the form:
Lim {Ax(f(a) + f(a + Ax) + f(a + 2Ax) +... f(a +n- 1Ax))}
(-x xAx for Ax==).
Employing the sign of the differential dx = Lim Ax, we follow
Leibnit z in denoting the sum by the abridged symbol:
x
IV. Jf(x)dx = Lim {Ax(f(a) + f(a + Ax) +. f(x- Ax))}
a
for Ax s —, 0 ( = x);
and call it the Definite Integral of the function f(x) taken from the
lower limit a to some determinate upper limit x.
The integral sign f is a sign of a sum; on the left side of the
defining equation IV. stands a symbol, on the right an expression that
can be calculated. It is to be observed regarding this formula, that
x as upper limit represents a definite value, but under the integral
sign it signifies a variable, since f is to be formed for the points
f(a), f(a + dx), f(a + 2dx), etc..
The conception of the integral as a sum gave rise to an erroneous
impression. For, if we first put Ax equal to zero on the right side
of equation IV., since all terms have the factor Ax, we obtain only
summands whose value is zero, and however many of them are added,
the resulting value of the sum is necessarily zero. An integral could
therefore never have any value but zero, or equation IV. should
contain a contradiction. This is not removed but only obscured by
"the further contradiction: f(x)dx is not zero but an infinitely small
quantity. Euler therefore (see foot-note ~ 105) completely rejected the
definition of the integral as a sum, and maintained only the definition
that follows from inversion of differentiation. Meanwhile, as the
above development shows, this same definition leads unavoidably to
the conception of a sum and this contains no contradiction, when
we bear in mind, that
Jf(x)dx
a
is not the sum of the limiting values of f(x)Ax, but the limiting
value of the sum of the terms f(x)Ax; in other words: what is
required is, first to find the sum for a finite number of terms as a
function of Ax, and then to determine its limiting value for Ax =-0.