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.

~ 37. 38.

Theorem of the Mean Value.

67

that immediately concerns us is to know, that in each case there is
a value for 0 between 0 and 1.*)
38. By the formulas just established the problem of actually calculating the values of a function for a given interval of its argument x is solved. Previously, with the exception of the process of
inclusion within limits, we had no means of doing this, even for the
elementary functions: x" (n arbitrary), ax, alog x (a positive), the
trigonometric and circular functions; and yet in its absence we could
perceive their uniqueness and continuity, and assign all their derived
functions in terms of the same symbols of calculation.
The only arithmetical operations we have in our power actually
to carry out are the two - summation and multiplication - performing
them a finite number of times on a definite set of rational numbers,
positive or negative, integer or fractional; irrational numbers must be
replaced by their nearest rational approximations.**) Accordingly the
only function whose calculation we can deem completed is the rational
algebraic function:
f( aoxn + a, xn-l +. a-1 x+ an
b-xm - bl I "-I +. b,,_x + b,
To calculate any other function for an arbitrary value of x, is to assign
a method according to which continued summations or multiplications
have to be carried out, the result of which exhibits the required value
with greater approximation the more frequently the operations are carried
out as prescribed by the method.
The elementary functions of x must therefore admit of expression
in the form either, of sums whose summands can be powers of the
argument x, or, of products whose factors can likewise contain the
argument in powers. When they are calculated in this manner, they
can themselves be used in the calculation of more complicated functions.
The number of such summands or of such factors will of course, il
analogy with the exposition of an irrational number, be infinite, for
otherwise every function could be- brought to the rational algebraic
form; but the arrangement of them will be such, that even a finite
summation or multiplication is enough to generate a value whose
difference from the required value of the function is demonstrably
*) The first formula was developed by Lagrange: Theorie des fonctions
analytiques, 1797; the alteration contained in the second was given by Cauchy:
Exercices de mathematiques, T. I. p. 29. Subsequently, still more general forms
for the last term were devised by Schlomilch according to the method we have
here followed.
**) Subtraction is summation with negative, division is multiplication with
fractional numbers.
5*