Measuring our Universe

In the 3 century BCE, the Greek mathematician Eratosthenes was able to estimate the Earth’s circumference. He learned that on the summer solstice the Sun at noon was reflected in a deep well in the southern Egyptian city of Syene, which indicated that the Sun was directly overhead at that time. He also knew the Sun was not directly overhead at that time where he lived (Alexandria, in northern Egypt) but was about 1/50 of a circle (about 7.2◦) below the zenith. Since Alexandria was about 550 miles north of Syene, then Eratosthenes estimated that the Earth’s circumference was about 550 ∗ 50 = 27500 miles, a size that is only about 10% higher than the actual circumference. A distance that required more mathematically sophisticated methods was derived during the 2 century BCE by the Greek mathematician Hipparchus. He applied his newly developed mathematics, now called trigonometry, to observations of a solar eclipse at two different sites to estimate the Earth-Moon distance in terms of the radius of the Earth. Figure 1 illustrates a simplified version of the geometry of these observations. It assumes that the Eclipse occurred at noon, the two sites were on a north-south line, and the Sun is infinitely distant. Correcting for these details won’t have much impact on the final answer.