MadSci Network: Science History

Re: How did the Greeks calculate the distance from Earth to the Moon?

Date: Tue May 16 10:35:00 2000
Posted By: Dan Patel, Undergraduate, Chemistry Major/Math Minor, University of Houston
Area of science: Science History
ID: 957933107.Sh

I have read that the Greeks used information from solar eclipses to 
calculate the distance to the moon.  A philosopher by the name of 
Aristarchus noted that the moon was just barely able to cover the sun 
during a total solar eclipse, meaning that the two have the same angular 
size, or apparent size, even though they are at different distances from 
the Earth.  This means that the actual sizes of the moon and the sun could 
be used to determine their distances by the use of proportions (see 
below).  Of course, for this to work you need to know the actual sizes of 
the sun and moon, and the distance of the earth to the sun (all of these 
were known to the Greeks).

distance to moon      actual size of moon
----------------  =   -------------------
distance to sun       actual size of sun

Another way the Greeks were able to determine the distance from the earth 
to the moon comes from a concept called parallax, which is a sort of 
triangulation.  This means that a stationary object appears to move if an 
observer changes his position.  You can test this idea by placing a pencil 
at arms length in front of your eyes and directly in front of your nose.  
Close your left eye and note the position of the pencil.  Now, at the same 
time open your left eye and close your right eye.  The pencil appears to 
have shifted position.  We can pretend that the moon is the pencil, and 
that each eye represents an observer.  If we can find the angle that the 
moon makes with one observer (one eye), we can use the that angle to find 
the distance of the moon using the following equation:

distance to the moon = (1/2 distance between observers) * (tangent of the 
angle between the observer and the moon)

This equation works if the moon makes the same angle with each observer as 
seen in the "illustration" below where "a" and "b" are observers and "m" 
is the moon.  This means that the moon and the observers make a huge 
isosceles triangle.  Of course, we can still do this if the observers see 
the moon at different angles.  It all has to do with the distance from one 
observer to a perpendicular line dropped from the moon multiplied by the 
tangent of the angle that the moon mmakes with that observer.  Below are 
some web sites with great illustrations on how to do this.

       * | *
      *  |  *
     *   |   *

The following site actually shows how the Greeks used parallax to 
calculate the distance to the moon and the sun. http://space

A fellow Mad Scientists posted an experiment one can do at home showing 
how to use the parallax concept. http:

The following web site shows how to use parallax to find the distance to 
the stars, but the same principles can be used to find the distance to the 
moon. http://zebu.uo

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