You've probably heard the old story that an eclipse was caused by a dragon eating the moon or sun, and that some human action or other would help drive off the beast. Here's actual photographic evidence! We have determined that the best way to drive off the dragon is to use our products. (The marketing department made me put that in) Seriously, the emotional impact of an eclipse is still high, especially a solar eclipse: there's even a tour industry which will take you to them.
Eventually, of course, people realized that eclipses were shadows cast as the sun, earth and moon came into a straight line (a syzygy, if you like weird technical terms)
The area on the right side of the earth in the picture is in night because it's in the
shadow cast by the earth. The shadow extends out into space behind the earth, just like your shadow behind you.
Because the sun is bigger than the earth, the shadow is actually shaped like an ice cream cone
(a triangle seen from the side). 
The shape of the cone is much longer and narrower than shown here. You
can measure that shape with any shadow-casting object, since the shape is always the same,
determined by the apparent size of the sun. The simplest way is to hold up a small disk and find at what
distance it just blocks the sun's disk, but this is an excellent way to
go blind, so DON'T
DO IT! Instead, take advantage of a quirk of the solar system: the full moon is the same visual
size as the sun (as is shown in solar eclipses, where the moon just about
covers the sun
...of course the sun is physically much bigger but also much further away). If you
safely measure the moon size this way, you'll find you have to hold the disk about 110 times
as far away as it is wide (for example a one inch disk would be nine feet away, or a half
centimeter disk at arm's length). So the earth's shadow extends out into space for 110 earth
diameters, too long to show in the picture.
(Note: Here is a safe
simple way to measure the sun
size directly. Here
is a writeup on measuring the moon size.)
This sort of scientific knowledge did not immediately drive out a superstitious
fear of eclipses: there's a famous story in Thucydides'
History of the Peloponnesian War
showing the power of eclipses. The Athenian army had invaded Sicily and was doing badly. The
leaders arranged an evacuation by ship, but just before it was scheduled, an eclipse of the moon
occurred. The army, including the leadership, was sufficiently unnerved that that they
consulted the gods for the meaning of the omen. Scientific explanation is all very nice,
but... the soothsayers told them the gods wanted them to
wait. They did, and they were defeated and captured. This was the Golden Age of Greece,
and Athens was the most powerful and intellectual city!
Everyone used numbers to describe the motions of the sun and moon; calendars were one of the first uses of mathematics. Naturally early scientists tried to predict eclipses using mathematical methods, but this is much harder than just a calendar. There is still a quantity in use for describing lunar motion called the draconitic month, the month of the dragon (Draco is the Greek word for dragon, as in the name of the constellation), because it's important in eclipse predictions.
The Greeks made progress in eclipse prediction, culminating in Ptolemy's Almagest, but along the way they found other surprising uses for eclipses. For instance, it's easy to actually compute how far away the moon is, by measuring what happened during a lunar eclipse. In particular what you need to know is the size of the shadow. (We've come a long way here...imagine seeing a dragon devouring the moon and your reaction is trying to measure exactly how big his mouth is!)
Next: Sizing the Dragon's Mouth
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