The Galileo's Moon Mountain Model illustrates the method used by Galileo to measure the height of a mountain on the Moon.  Using his improved telescope design, Galileo was able to see spots of light in the otherwise dark potion of the Moon.  He interpreted these spots as mountain peaks which caught the rays of the sun even though the sun did not illuminate the Moon's surface at the base of the mountain.  He measured the distance of the bright spot from the terminator (the line separating the lit and unlit portions of the Moon) as a fraction of the Moon's radius.  Then he was able to use a geometrical argument to determine the height of the mountain as a fraction of the Moon's radius.  Galileo knew that the Moon's radius was approximately 1600 km (he didn't use those units, of course), which allowed him to determine the absolute height of the mountain.  (Note that the modern value for the Moon's radius is about 1740 km.)

One window shows the view from above the North pole of the Moon.  The mountain appears near the bottom of this window.  A ray of sunlight which just grazes the Moon's surface at the terminator is shown. Controls allow the user to adjust the angle of sunlight (thus altering the Moon's phase) and the height of the mountain.

The other window shows the view from Earth.  When sunlight strikes the top of the mountain a bright spot becomes visible in the dark area of the Moon. Likewise, when the mountain is in the bright region it casts a shadow.  The distance across the Moon's face from terminator to mountain in shown.

Please note that this resource requires at least version 1.5 of Java (JRE).