Distance to the Moon

Aristarchus was the first known person to calculate the distance to the moon with reasonable accuracy.
To do this he needed the following pieces of information:
1.  The diameter of the earth.
2.  The length of the earth's umbra
3.  The relative diameter of the moon to the earth (remember that Aristarchus used a lunar eclipse for this).  We are assuming for these calculations that the earth's umbra is parallel.  Is this a fair assumption?  Why or why not?
4.  The fact that the moon's umbra ends on earth  (How do we know this?  What evidence did Aristarchus have?)
5.  Using the above information we can create similar triangles and solve for the unknown (the length of the moon's umbra, or the distance from the earth to the moon).

I expect that you will have a drawing in your book that describes this, but that you will also have a write-up that describes the process and how Aristarchus figured this out.  Let me know if you need more clarification with this.

Mr. H