Thousands of years ago, ancient astronomers like **Aristarchus** and **Eratosthenes **combined careful observations with simple mathematics to measure the **solar system, **especially the diameters *D* of **Earth**, **Luna** (Earth’s moon), **Sol** (Earth’s star, the sun), and the radii *r* of their orbits. You too can do this, but it helps to observe an **eclipse** or two.

##### Step 1: Diameter of Earth

Measure how shadow lengths vary with latitude. No need to pace the distance between **Alexandria** and **Syene**, just use your favorite map software! Assume Sol is far from Earth (and check in Step 3), so Sol’s rays are nearly parallel.

##### Step 2: Distance to Luna

Measure the duration of a **lunar eclipse**.

##### Step 3: Distance to Sol

Measure the angle between Sol and Luna at first quarter moon, when Luna appears to be ahead of Earth in its orbit, like a signpost to a car on a road. This is the most difficult step, as the angle is nearly but not quite ninety degrees, but the result is the **astronomical** **unit**. (Alternately, measure the ratio of time Luna is **crescent** to **gibbous**.)

##### Step 4: Diameter of Luna

Measure the time for Luna to enter Earth’s shadow during a lunar eclipse. Consistent with Step 3, again assume Earth is far from Sol, so Earth’s shadow is nearly uniform.

##### Step 5: Diameter of Sol

Note that Sol and Luna have about the same apparent angular size (both subtending about half a degree). This is most spectacularly evident during a **solar eclipse**, where Luna just barely covers Sol — if you were ever fortunate enough to experience a solar eclipse.

Thanks, Mark! I enjoy reading your posts as well.