Download Measuring Earth`s Circumference with a Rod

Measuring Earth’s Circumference with a Rod In your hand you hold a rod. Your task is to measure the circumference of the earth. How would
you go about it? This Herculean feat was accomplished by a Greek mathematician and scientist
called Eratosthenes, who pulled it off some time around the year 200 B.C. using an astonishingly
simple application of geometry.
It’s all about transverse angles. Transverse
angles occur when a line crosses two or more
lines in the same plane. If a line crosses parallel
lines in the same plane, many of the resulting
angles will be congruent. The angles with
matching colors in the adjacent diagram are
congruent angles that result when a line l
crosses parallel lines q and r. Angles 1, 4, 5,
and 8 are congruent, and angles 2, 3, 6, and 7 are congruent.
Armed with the basic facts of geometry, a rod, and the latest cutting edge technology for
measuring angles that could be found in 200 B.C., Eratosthenes set out to calculate the
circumference of the earth.
But wait a minute. How did he know the earth
was round to begin with? Good question. One of
the reasons Eratosthenes concluded that the earth
was round was because of shadows. He reasoned
correctly that because the sun was so far from the
earth, all the rays of the sun would be hitting the
earth at approximately the same angle at a given
moment. If the earth were flat, objects of the
same height would cast shadows of the same
length everywhere on earth at precisely the same
time.
But Eratosthenes knew that shadow
lengths actually differed by
comparing them in various cities at
the same time of day. For example, he
knew that at midday on June 21st in
the city of Syene, Egypt there were no
shadows. Tradition has it he could see
the reflection of the sun in a deep
well-shaft, which implied that the sun
was directly overhead. But in the city
of Alexandria, Egypt, which was
about 500 miles north of Syene, there
were shadows on June 21st at midday.
The difference in the length of the
shadows convinced Eratosthenes, as well as other scholars, that the earth must be round, not flat.
Looking at the exaggerated diagram, you may notice a similarity to the transverse angles we
considered earlier. Since the rays of the sun are parallel, a rod pointing toward the center of the
earth at Alexandria could form a line crossing parallel lines, and the result would be transverse
angles.
That’s where the rod came in handy. Eratosthenes placed a rod perpendicular to the surface of
the earth in the city of Alexandria. Since it was perpendicular, it pointed toward the center of the
earth. On June 21st at midday, Eratosthenes measured the angles of the triangle formed by
joining the tip of the rod with the end of the shadow. The lower angle was found to be 82.8
degrees, and the upper angle was 7.2 degrees.
Using his knowledge of transverse angles, Eratosthenes knew the upper angle of the triangle
formed by the rod and the shadow would be congruent to the angle at the center of the earth,
which would therefore be 7.2 degrees. He also knew the distance between Syene and Alexandria
was approximately 500 miles. Everyone knows that a circle has 360 degrees. That was all the
information he needed to make his calculation.
A simple proportion will do the trick. We use the fact that 7.2 degrees is to 500 miles as 360
degrees is to the circumference of the earth. Letting C represent the earth’s circumference, we
could set it up like this:
7.2 360
=
500
C
7.2C = (360)(500) Cross multiply
C=
(360)(500)
Divide both sides of the equation by 7.2
7.2
C = 25,000 miles
So earth’s circumference was estimated to be about 25,000 miles.
7.2
1
1
is the same as , so 500 miles is
of the circumference of the
360
50
50
earth. In an equation, it looks like this:
1
500 =
C Multiply both sides of the equation by 50
50
(50)(500)=C
We could also reason that
C=25,000 miles
And that’s the story of how Eratosthenes used a simple geometry concept, a rod, and a great deal
of creativity to produce a remarkably accurate estimate of earth’s circumference for the first time
in history. A modern estimate of earth’s polar circumference is 24,860 miles. The difference of
140 miles between the two approximations is very small. However, during the time period when
Eratosthenes made his calculation, two different lengths of stadium were commonly used for
measurement: the surveyor stadium and the smaller Olympian stadium. The estimate of 25,000
miles is based on the surveyor stadium and is far more accurate than an estimate based on the
Olympian stadium. While scholars are still debating which unit of measure Eratosthenes used, I
say we give the inspiring old wizard the benefit of the doubt.
By Elaine Califf