![]() ![]() Remember, you must drop it down in a straight line that is plumb, without moving it back-and-forth. Without moving the pen, pull your desk lamp down to within 3 inches from the top of the pen. Take a pen and move it off of dead center (plumb) until you get a shadow that is 2 inches. 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). ![]() 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. 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). ![]() 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. Hence, $\triangle OSM$ is an acute, nearly isosceles right triangle, whose legs are practically parallel and much, much greater in length than the base. 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 $\angle OMS \approx 90°$. The sun is in the direction of the horizon, and the 1st-quarter moon is near zenith, hence $\angle SOM \approx 90°.$ $ The only angle the observer can measure directly is of course the angle between the sun and moon, the observer forming the vertex. 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.įorm a triangle out the observer, the sun and the moon: $\triangle OSM. 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. ![]()
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