Calculus: Analytic Geometry and Calculus, with Vectors

7.1 Curves and lengths 413

the earth is much like a spherical ball, but is not a spherical ball because moun-
tains and valleys are quite noticeable around the Mediterranean sea and some
other places. To show that a little information about lengths of arcs can have
quite astonishing consequences, we look at the method said to have been used
by the industrious Eratosthenes (c. 275 a.c. to c. 195 a.c.) to find what the radius
(a number) of the earth would be if the earth were sandpapered to the shape of a
spherical ball. Figure 7.194 shows 0, the center of the earth,
and a circular arc 11B on the surface of the polished earth.
The dotted vertical lines represent rays of light coming from
the sun, and these rays are so nearly parallel that approxi- A
mations can be based on a figure in which the rays are parallel
to the line 011. Thus an observer at B finds that the sun is
at his zenith. At B a pole BC of height h is erected in such
a way that an observer at B thinks it is vertical, that is, the
points 0, B, C lie on the same line. In addition to h, two
other lengths are obtained. In the first place, we find the
length a of the circular arc DC which has its center at B (the 0
base of the pole BC) and which has a shadow that just covers Figure 7.194
the pole. In the second place, the length b of the arc 11B is
obtained by more or less reliable surveyors. Let r be the radius of the earth.
Since parallelism of the light rays implies that the angles DBC and ,40B are equal,
say to 0, we obtain the equation

a b

h r

from the fact that each of the ratios is equal to 0. From this equation, r is easily
calculated. It is not to be presumed that Eratosthenes spoke English and knew
about miles, meters, and radians, but it is said that his calculations produced
estimates of the radius and circumference of the earth that are almost as good as
the estimates (radius 4000 miles, circumference 25,000 miles) that are ordinarily
used for rough calculations. The fact that reliability of computed results depends
upon accurate surveying is precisely the reason why two men named Mason
and Dixon were commissioned to do some accurate surveying.
10 This problem, like some others, does not have a number for an answer.
It requires us to think about connections between graphs and curves, and to
learn some geometric terminology. The graph of a polynomial of degree n,
that is, the graph of an equation of the form

(1) y = ao + a1x + a2x2 + ... + a,,xn,

in which an 7-1 0, used to be (and occasionally still is) called a parabola of degree n.

For example, a line is a parabola of degree 1 and a cubic is a parabola of degree 3.

While this terminology is almost extinct, the graph of the equation

(2) y2 = x3

still is called a sem.icubical parabola, the old idea being that, when we solve for y,

we get an exponent which is not an integer but is half of 3. According to this