The Extraordinary Eratosthenes and the Amazing Aristarchus

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The Extraordinary Eratosthenes and the Amazing Aristarchus
Long before Magellan sailed around the Earth in 1492, two extraordinary Greek
Philosophers had already determined that the Earth was a sphere, a close estimate of its
circumference and diameter, the Moons size relative to the Earth, and reasonable relative
distances to the Moon and Sun. Here’s how they did it well over 2000 years ago.
About 200 B.C., Eratosthenes, the librarian of the Great Alexandria library, determined
the circumference of the Earth by noting the Suns rays falling on a stick stuck 90° to the
ground in Alexandria casting a shadow, while at the same time a stick in Syene a known
distance away cast no shadow at the same time of day. How did he do it? Look at the
picture below and try the math yourself.
Eratosthenes measured angle A, the
angle of the stick to the end of its
shadow. Angle A = 7°. Recall that
angle A & angle B are complementary,
so angle A = angle B. He knew the
distance between the two sticks was
785 km. The circumference of the
Earth can now be found
proportionally: (show your work!)
7
360
=
Solve for x. x =
785
x
1. Was your answer close to the actual approximation of the Earths circumference of
40,000 km?
2. Explain how Eratosthenes logic for determining the circumference of the Earth was
based on the earlier Greek mathematical discovery that a circle is equal to 360 degrees.
Now Eratosthenes could find the relative size of the Moon. During a total lunar eclipse
the Moon passes through the Earth’s shadow. By timing the passage of the Moon
through the Shadow Eratosthenes found his
answer. Try it yourself below.
Eratosthenes measured the time from when the
Moon touched the Earths shadow until it was
entirely within the shadow. T1 = 50 min.
Then he timed how long it took for the Moon to
move through the shadow. T2 = 200 min.
50 min .
=
200 min .
Did you find the Moon was 0.25 the size of the
Earth? Now find the actual size of the Moon.
First find the diameter of the Earth and then the
diameter of the Moon. Use the formula
Circumference = Diameter x π.
Assume Earth C = 40,000 km.
Show your work!
Did you find the Earth’s circumference equal to about 12,700 km and the Moon’s
diameter about 3,200 km? Now Eratosthenes could find the rough distance to the Moon.
By holding out his arm he could just cover the Moon in
the sky with his thumbnail. He measured the distance
from his eye to his thumbnail as about 100 times the width
of his thumbnail. The distance to the Moon was
proportional to this measure. Try it yourself.
1
3,200
=
Solve for x. Show your work.
100
x
x=
An earlier Philosopher, Aristarchus in 300 B.C. estimated the distance to the Sun using
trigonometry (the study of right triangles). Try it yourself below.
Aristarchus assumed that when the
Moon was half full that it must be at
right angle (90°) with respect to the
Earth and Sun. He measured MoonEarth-Sun angle as 87°. Knowing that
all angles in a triangle add up to 180°,
the Earth-Sun-Moon angle was:
180° – 87° – 90° = 3°
Without going into the mathematics of
trigonometry try the equation below.
The ratio of the Moon-Earth-Sun angle is 0.05 and is equal to
0.05 =
MoonDist
so…
SunDist
MoonDist
Solve for the Sun Distance assuming a Moon Distance of 320,000 km
SunDist
Show your work!
SunDist
=
MoonDist
Did you find the Sun about 20 times farther than the Moon? In fact the Sun is about 400
times further than the Moon because the correct measurement for the Moon-Earth-Sun
angle is 89.85 and Aristarchus could not measure this angle precisely enough given the
technology of his day. But the point is these early Greek Philosophers were using logical
methods and mathematical techniques they had developed to begin exploring their
Universe well before this knowledge was lost in the middle ages!
Now find how far the Sun is relative to the Moon:
4. Explain below why the discoveries of Greek mathematics were crucial to the work of
Eratosthenes and Aristarchus!
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