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.

20

Functions of a variable.

Bk. I. ch. IV.

the form f (x, y) = 0 where f is integer and rational in x and y, we
call y an algebraic function of x. Accordingly the most general type
of an algebraic function with two variables is
Aoyn + Alyn-1 + A2y —2 +... +- An-_y + An - 0,
where the quantities A are polynomials of any degree in x of the form
A =- aoxm + aIxm-1.. + am,, n and n being positive integers.
Here the only arithmetical operations performed on the variables
are those of the first four species repeated a finite number of times
and including integer powers.
All other functions are transcendental. To these the operations
of arithmetic contribute the power with an irrational exponent, but
chief of all the exponential function, the simplest type of which is
y = ax, and the logarithm y =- log x; we can only calculate these last
two functions, so far as our conception of number yet reaches, for
positive values of a and b, and further the second only for positive
values of x.
12. Not less important are the trigonometric or as they are
sometimes more precisely called goniometric functions which we are
familiar with from the elements of geometry. These are geometrically
defined as the ratios of lengths which depend on an angle. In order
to indicate that angular quantity is always a pure number, we describe
the magnitude of an angle, not by degrees, but by the ratio of the
length of the circular arc to that of the generating radius which belongs to it. The length of the circular arc belonging to an angle, as
well as that of the whole circumference, is proportional to the radius
with which it is described. Hence the circumference may be denoted
by 2rr, where z is a number to which the Geometry of Euclid shows
we can approximate as closely as we please by inscribing and circumscribing polygons. Accordingly, to each angle there will belong a
number which determines as part or multiple of 2x the arc as part
or multiple of the circumference. By geometric investigation we conclude the following properties of the functions:
y = sinx and y = cos x.
1) as x goes from 0 to rz, sinx goes from 0 to 1, cosx from 1 to 0., x,,,, -, sinx,, 1,, 0, cosx, 0,,-1,, x,,,7, Z.tC, sinx,,,, 0,,-1, cosx, -1,, 0
Z 2 z sinX I Cos,.
1) x.,, -,,2 sinx,,,, -1,, O cos x,, 0, 1.
This is one of the cases in which it is good to adopt a distinction
of signs in the geometric interpretation.
2) (sinX)2 + (cosX)2 - 1.