8.1 Trigonometric functions and their derivatives 439lengthand provetohim that the arc
IVQ(h a 2 _h)
has alengthwhichwe may denote by s. I
Knowing that our quantities depend
upon h and a, we can define a number 8
by the formula
s length of arc - v1 h A(a.O) x
(812) 8 = _ -a radius Figure 8.11If we multiply each of a and h by the same positive constant X (lambda),
we get a new radius and a new arc, but the ratio s/a remains the same.
To prove this, we must prove that the length of the newarc is the product
of X and the length of the original arc. Our theory of length enables us
to do this because, by drawing radial lines from the origin, we can see
that each polygon inscribed in the new arc determines (and is determined)
by a polygon inscribed in the original arc and that the lengths of the
straight segments of the polygons all differ by the same factor X. All
this shows that we get the same B when we take another pair of values of
a and h for which the point (h, a2 - h) lies on the same half-line extend-
ing from 0 through Q. Now we can again simplify (or complicate)
matters by considering the number 8 to be "a measure of the amount of
rotation required to bring a line from the position 04 to the position
OQ" or "a measure of the opening between the lines 04 and OQ." Per-
haps to remind us where the number 9 came from, or perhaps to indicate
something of which 0 is to be considered a measure, we complete Figure
8.11 by inserting the 9 together with the curved arrow which shows the
direction of our arc. It is the fashion to call 8 an angle, but it should be
permanently remembered that 0, like 5, is a number. The facts that we
sometimes use 0 to measure an amount of rotation and use 5 to measure
a number of fingers do not imply that 8 is a rotation and that 5 is a fist-
ful of fingers, but we can nevertheless understand and even use the more
or less convenient terminologies involving "angles" that have become a
part of nonscientific as well as scientific attempts to convey information.
If all this indicates that trigonometry is a subject much too difficult for
inclusion in trigonometry textbooks, we do have one consolation. The
hard work is done and the rest is easy.
Our theory of curves is sufficiently general to allow us to extend the
above account of positive (that is, counterclockwise) arcs and angles to
cover situations in which the arc is longer and Q lies in the second or
third or fourth quadrant. Moreover, the arc can be so long that we
must encircle the origin more than once to traverse it, and the number
(or angle) 0 is still defined by the same formula (8.12). In case the
arc starts at A and is oriented in the negative (or clockwise) direction,
everything is the same except that the directions of the arrows are
reversed, a negative sign is prefixed to the middle and last members of