Never Mind the Bullocks, Here's the Science (12 page)

Now the Moon and the Sun look the same size in the sky. So if you look at the Sun, you are again setting up a triangle with the height:base ratio of 108:1.

Just once more—any time you look at the Sun or the Moon, you set up a triangle with a height:base ratio of 108:1.

At this stage, we have to learn a tiny bit about Similar Triangles. In Similar Triangles, corresponding sides are in the same ratio to each other. In the pic below, ABC and EFG are Similar Triangles (because each side in EFG is twice as big as the corresponding side in ABC).

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Triangles are similar if they have the same ‘shape’, but not necessarily the same size. You can think of it as ‘zooming in’ or ‘out’, making the triangle bigger or smaller, but keeping its basic shape.

Here’s another pair of Similar Triangles—ABC (the big one) and AEF (the little one)

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ABC and AEF are Similar Triangles.

The Sun is a lot bigger than the Earth. So the shadow that the Sun casts behind the Earth has the shape of a cone.

The Sun Is Bigger Than the Earth

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The Earth casts a shadow in the shape of a cone.

If you stand at
A
(which would mean you were mysteriously floating in space), then the Earth (
B C
) will exactly block the Sun.

This means (from pic 2) that
  

A is the tip of the cone of the shadow of Earth thrown by the Sun. If you stand at A and look at the Earth as it exactly blocks the Sun, the triangle ABC has the height:base ratio of 108:1.

(If you want to be super accurate, the true height of triangle ABC should be measured from the centre of the line, BC. But ABC is a very skinny triangle, so AC is close enough to the true height.)

Eratosthenes measured that the circumference of the Earth is about 40,000 km.

The Ancient Greeks knew that pi, the ratio of the circumference to the diameter, is about 3.14.

So the diameter of the Earth is about 12,700 km.

Back to Eratosthenes

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The circumference of Earth = 40,000 km

= 3.14 = ratio of the diameter to the circumference

Therefore, the diameter of the Earth

We already know that the triangle ABC has a height:base ratio of 108:1.

Length of Shadow Cone

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Triangle ABC has a ratio of 108:1 (from pic 5)

Therefore,

But
B C
= diameter of the Earth = 12,700 km

Therefore
A C
= 108 × BC

                      = 108 × 12,700

                      = 1,372,000 km

So the distance from Earth to the end of the cone of shadow (point A) is about 1,372,000 km.

We now need a person with a clock—a bucket with a small hole in the bottom is good enough, so long as you keep topping up the water to keep it at roughly the same level.

During a lunar eclipse, the Moon moves into, and then out of, the conical shadow of the Earth. It takes about 4 hours (on our modern clocks) for the Moon to cross the shadow. As the Moon cuts into and out of the Earth’s shadow, it gets darker and changes colour.

Now count the time between when the Moon first kisses the shadow, and when it is just fully inside—say, 100 drops of water.

Now count the time between when the Moon first kisses the shadow, and when it has just fully left the shadow—say, 250 drops of water.

How Big is the Moon Compared to the Width of Earth’s Shadow?

So the width of the Earth’s shadow (at EF) is about 2.5 times the diameter of the Moon.

Now it’s time to play Similar Triangles.

Triangle ABC is a 108:1 triangle (because you are looking at the Earth exactly blocking out the Sun—from pic 5).

Triangle AEF is a 108:1 triangle (because it is similar to triangle ABC—from pic 4).

Triangle CDF is a 108:1 triangle (because any time you look at the Sun or the Moon, you make a 108:1 triangle—from pic 2).

Lunar Eclipse + Similar Triangles

(i) The Moon travels upward through the cone of shadow on the dark side of the Earth.

(ii)
A B C
is a 108:1 triangle (from pic 5)