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.

242 The definite integral as the limiting value of a sum. Bk. III. ch, VI.
~t = v,-=nv
At-1 =1
and denote their values by Av and ~B. In the series of'numbers:
A1, A2, A3,.. A..., each number is less than, or at most equal
to, the preceding, for, whereas one interval ex. gr. d2(1) contributes
d2(') G2(1) in A1, the same interval in the sum A2 consists of several
parts. But these partial intervals certainly do not contribute more
to the sum A2 than the product dC2(1)G2(1), because G2() denotes the
greatest number that occurs in the entire interval d2(1), and therefore
also in any of its subdivisions.
The quantities.BlB2, B3,....B.. form a series of increasing
numbers; and since each A is greater than each B, the series of
numbers A as well as the series B has each a definite limiting value.
These limiting values become identical when a place v can be assigned
for which and for all greater values of v the difference A, - Rv is
less than an arbitrarily small number a, i.e. such that:
/U — n, yt = nV
Av - B - (v) (Gl() - gye(T)) - 7dg()DYD(V) < 6.
/ —=1 ^ -=1
In whatever way the quantities 0 are assumed, the sum:
S- = dl(f)f(a + - O dj(")) +- d2(')f(a +- c() +- 02()d2(V)) +..
+- dnv" f(b --,n In. ))
whose limiting value defines the integral, always lies between the limiting
values of A, and B, and when these are equal, this sum has also the
same finite and determinate value; as we undertook to prove.
The condition enunciated is sufficient; but it is also necessary.
For if both series of numbers A and B had not the same limiting
value, the limiting value of the integral sum could, by varying the
quantities 0, be brought to coincide with the limiting value either
of A or of B; thus it would not be independent of the quantities 0.
But the proof is not yet complete; for it was assumed, that the
successive partitions are always carried out so that the extremities of
a partial interval occur also as extremities in the subsequent partitions.
The questions therefore arise: Is the value of S quite independent
of the choice of dividing points? and is it also independent of the
manner of continuing the partition? Let:
a,) x, X27 ~. ~ Xn —p b
and:
a, X ', X2,. o X'm-a b
be the extremities in two partitions. Suppose these in whatever way
commenced to have been carried on quite independently according to