Subject: How did Cassini establish the earth-sun distance?

First, I assume you have read the explanation I gave about what Huygens did. In that I mention that Cassini used Mars to find the length of the astronomical unit (also called an AU, the distance of the Earth to the Sun).

This had been the Holy Grail of astronomy for a long time. Using geometric arguments, it had been long possible to get the relative distances of all the planets. For example, it was known that Jupiter was 4 AU away when it was at its closest to the Earth. But what was an AU?

Enter Gian Domenico Cassini. He sent another astronomer, Jean Richer to Cayenne, French Guiana in South America. They observed Mars at the same time, carefully measuring its position in the sky relative to background stars. Since the two astronomers were located at different point on the Earth, they saw Mars in slightly different positions. This effect is called parallax, and I have a detailed description of it on my own website.

So, knowing the distance between the two observing points, they were able to calculate the actual distance to Mars in miles, which in turn they were able to convert to AU. This was a major step in our learning the distance scale of the entire Universe: we use the AU and parallax to find the distance to nearby stars! I wonder if Cassini knew just how important this work was...

By the way, the measurement Cassini and Richer made was very difficult to do, and their number wasn't terribly accurate. However, it was the first big step, and the number was later refined. Now we know it to an accuracy of meters!

I found out the answer to this question at the Mars Timeline of Discovery webpage and the Astronomy Online website.