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.

Eighth Chapter.
General theorems concerning the double integral.
166. Definition of the double integral.
Let f(x, y) be a function of the two variables, that is uniquely
defined in any way for any domain T, but for the present in such
a way as to be everywhere finite. Let the domain be conceived
to be in the plane xy surrounded by some continuous and closed
succession of points, or, stated analytically, bounded by a curve whose
equation is p (x, y) = 0. The domain can also be bounded by more
closed lines than one, as ex. gr. a circular ring by two circles. The
simplest case of boundary of a domain is a rectangle with its sides
parallel to the axes of rectangular coordinates; then x takes all values
from a to b, y all values from a to 3. Should the function be defined
for the entire infinite plane, we can always express this: it is defined
for a surface, whose boundary can be arbitrarily extended. Let us
resolve the domain T, at first on the hypothesis that it is
finite, into n small parts or superficial elements, and call them rl,
r,2... n. All these elements are conceived as positive quantities.
Such a resolution is effected, ex. gr., when we cover over the domain
with a net having its lines parallel to the coordinate axes at the
distances Ax and Ay. In this case all superficial elements are equal,
being rectangles whose magnitude is Ax. Ay. Only at the bounds
of the domain are these rectangles cut by the boundary line. Let us
select any arbitrary value among those assumed by the function within
or at the limits of such a superficial element. For simplicity, let
such a value in each be denoted by /'(z^), f(r)),..(. '(,,); thus the
question arises:
Under zwhat hypotheses does the value of the sum:
Sn =f(Q,).,t + f(r,,) + * + f() * T,
approximate to a determinate limit, altogether independent of the choice
both of the superficial elements and of the value of the function in any
szch element, when the number of the elements is arbitrarily increased
according to any law in such7 a wvay that each element tends to the'
limit zero?