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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?

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up vote 7 down vote accepted

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.

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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).

enter image description here

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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. – WetSavannaAnimal aka Rod Vance Oct 23 '13 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. =) – David H Oct 23 '13 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. – David H Oct 23 '13 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. – Jack M Jun 20 '14 at 9:55

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