# How did Eratosthenes know the suns rays are parallel?

Eratosthenes famously observed that the suns rays were perpendicular to the ground in one location, yet non-perpendicular to the ground at a location some miles to the north. On the assumption that the sun's rays are all parallel, this means the earth is round (or at least not flat).

But how do we know the sun's rays are parallel? The rays of light from my desk lamp aren't parallel by the time they reach my desk. The spot directly below the lamp is hit perpendicularly, but the edge of the desk is hit obliquely. The issue is clearly determining whether or not the sun is far enough away that its rays can be assumed to be parallel. But how can we know that?

There's actually at least one very big clue that's been accessible to skygazers since the earliest times: the first quarter moon at dusk. Every child in the northern hemisphere going back to 30,000 BCE likely would have been familiar with how 1st-quarter moons always tend to rise at noon, reach its highest point at sunset (with an azimuth directly south), and set at midnight.

Form a triangle out the observer, the sun and the moon: $\triangle OSM .$ The only angle the observer can measure directly is of course the angle between the sun and moon, the observer forming the vertex. The sun is in the direction of the horizon, and the 1st-quarter moon is near zenith, hence $\angle SOM \approx 90°.$

The angle with vertex at the moon, $\angle OMS$, couldn't be measured in general, but it doesn't take too much imagination to infer that the shape of the sunlit portion of a 1st-quarter moon results whenever $\angle OMS \approx 90°$. Hence, $\triangle OSM$ is an acute, nearly isosceles right triangle, whose legs are practically parallel and much, much greater in length than the base. This small base length is the earth-moon distance $|OM|$ is itself much greater than any terrestrial distances we measure on the Earth's surface. Thus, with extremely little effort we can be reasonably confident that Eratosthenes' condition of parallel sunlight rays holds to good enough approximation for the purpose of his measurements (uncertainties in the measurements of distances between cities would have been the limiting factor towards overall precision anyway).

• Fantasic answer. But (although this is likely beyond the scope of this discussion), do we have any historical evidence that Eratosthenes reasoned like this? I should have thought it equally plausible that he simply assumed the Sun was very far away. Oct 23, 2013 at 7:48
• Yes, we do! The reasoning is actually original to Aristarchus of Samos (born ~35 years before Eratosthenes), and his fantastic text, On the Sizes and Distances of the Sun and the Moon. Read it. It's wonderful. =) Oct 23, 2013 at 8:03
• @WetSavannaAnimalakaRodVance I suppose I should qualify though that I'm not certain through what route Aristarchus' writings came to Eratosthenes. My best guess would be in letters with Archimedes. Oct 23, 2013 at 8:18
• Aristarchus of Samos: The Ancient Copernicus by Thomas Heath contains the only translation of On the Sizes and Distances I could find, for future reference. Jun 20, 2014 at 9:55

If one assumes that the Earth is flat and the angles of the shadows of two separated vertical poles are analysed, one might falsely calculate the distance to the sun instead of the curvature of the Earth.

But if a third pole is introduced it would not agree with the other two and that of the calculated sun distance. If more than two shadows were analysed, the curvature of the Earth can be proved.

I have no idea whether Eratosthenese did this or not.

• Can you share link on how you would use 3 poles proves curvature of earth? And does this proof work regardless of distance from Sun ie rays are parallel or not?
– Bob
May 13, 2022 at 21:08

To use the method that Eratosthenes used, I believe The results would be the same if the sun was 3000 miles or 93,000,000 miles away. Here is an easy way to prove it: With your desk lamp or any single filament light source (not a fluorescent light), place it three or more feet above your surface (the more distance, the better). Take a pen and move it off of dead center (plumb) until you get a shadow that is 2 inches. Without moving the pen, pull your desk lamp down to within 3 inches from the top of the pen. Remember, you must drop it down in a straight line that is plumb, without moving it back-and-forth. Notice how slowly the length of the shadow changes at first. As you get closer, notice how quickly the length of the shadow changes from two foot to three inches. Now, imagine the same test would be done on a streetlamp (or any single filament light source) and you raised the pen slowly to meet it. Did the length of the shadow change when you raised it 3 foot? Not at all! Imagine the difference between being 3000 miles or 93,000,000 miles away with a single three foot pole. SO, Eratosthenes was RIGHT!