How High The Moon? - Galileo And Einstein

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Michael Fowler, UVa Physics

In this lecture, we shall show how the Greeks made the first real measurements of astronomical distances: the size of the earth and the distance to the moon, both determined quite accurately, and the distance to the sun, where their best estimate fell short by a factor of two.

How big is the Earth?

The first reasonably good measurement of the earth’s size was done by Eratosthenes, a Greek who lived in Alexandria, Egypt, in the third century B.C. He knew that far to the south, in the town of Syene (present-day Aswan, where there is now a huge dam on the Nile) there was a deep well and at midday on June 21, the sunlight reflected off the water far down in this well, something that happened on no other day of the year. The point was that the sun was exactly vertically overhead at that time, and at no other time in the year. Eratosthenes also knew that the sun was never vertically overhead in Alexandria, the closest it got was on June 21, when it was off by an angle he found to be about 7.2 degrees, by measuring the shadow of a vertical stick.

The distance from Alexandria to Syene was measured at 5,000 stades (a stade being 500 feet), almost exactly due south. From this, and the difference in the angle of sunlight at midday on June 21, Eratosthenes was able to figure out how far it would be to go completely around the earth.

Of course, Eratosthenes fully recognized that the Earth is spherical in shape, and that “vertically downwards” anywhere on the surface just means the direction towards the center from that point. Thus two vertical sticks, one at Alexandria and one at Syene, were not really parallel. On the other hand, the rays of sunlight falling at the two places were parallel. Therefore, if the sun’s rays were parallel to a vertical stick at Syene (so it had no shadow) the angle they made with the stick at Alexandria was the same as how far around the Earth, in degrees, Alexandria was from Syene.

According to the Greek historian Cleomedes, Eratosthenes measured the angle between the sunlight and the stick at midday in midsummer in Alexandria to be 7.2 degrees, or one-fiftieth of a complete circle. It is evident on drawing a picture of this that this is the same angle as that between Alexandria and Syene as seen from the center of the earth, so the distance between them, the 5,000 stades, must be one-fiftieth of the distance around the earth, which is therefore equal to 250,000 stades, about 23,300 miles. The correct answer is about 25,000 miles, and in fact Eratosthenes may have been closer than we have stated here---we’re not quite sure how far a stade was, and some scholars claim it was about 520 feet, which would put him even closer.