Lunar Eclipse

A lunar eclipse is when the moon travels into the earth's shadow. Aristarchus believed that in the longest lunar eclipse possible, two moons fit into the earth's shadow.

This is a Lunar Eclipse.

The equation Aristarchus started with is above.

Earlier in the process we learned that ES = 19EM and Rs = 19Rm

Add EM and 19EM which becomes 20EM. Cancel the Rm.

After cross multiplying you will get this equation. Then subtract both sides by to 2x.

Now get x by itself by dividing by 17.

Now you know that x=40EM over 17.

He got this equation by using the smaller triangles. Now cross multiply.

Earlier we learned that x=40EM over 17. So now substitute 40EM over 17 for x.

After substituting 40EM over 17 you multiply it through.

Now simplify by adding all like terms.

Now solve for Rm by dividing both sides by 114EM.

Which leaves you with this final equation.

By Nate, Steve, Brian