The cosmic distance ladder is the sequence of measurements, insights and experiments that have been performed over the past two millenia to sequentially determine the distance and sizes of celestial objects. The ladder begins at determining the size of Earth and ends at calculating the size of the observable universe—as far as we know. This article is inspired by 3Blue1Brown’s video series with Terrence Tao.
A visitor looking down a well in Syene, Egypt, during the summer solstice around 300 BCE, noticed that the sun was perfectly centered in the well’s reflection. This was the visitor’s last day in Syene, so he figured he’d try the same experiment at home in Alexandria. He waited until noon during the following summer solstice and looked down the well again. Unfortunately for our main character, Eratosthenes, there was no sun in the reflection. Why? Being a learned man, he was aware of the Earth’s roundness and figured that the sun was not directly overhead. Using a gnomon—a sort of ancient protractor—to measure the angle of the sun in the sky, Eratosthenes calculated that the sun was off from being directly overhead by 7 degrees. Because he also knew that many miles directly south the same phenomenon was occurring in Syene, he figured that the distance between Alexandria and Syene was similarly 7 degrees of the Earth’s circumference. Professional bematists (people paid to walk with even steps and count them in order to measure distance) were hired, and found the distance between the two cities to 5000 stadia, a unit of measurement estimated to be equivalent to 157.7 meters. It's simple to calculate the ratio between 7 and 360 degrees: 51.4, thus multiplying the two gives 257,000 stadia, giving us around 40,500 kilometers for the circumference of the Earth. That’s only incorrect by 1.2%!
You, clever Reedie, might have two objections: How did Eratosthenes know the Earth is round, and how did he know it was the summer solstice? Aristotle had spent time observing lunar eclipses and noticed that the Earth’s shadow on the moon is consistently circular. Importantly, he knew that the only object whose shadow is always circular is a sphere. Thus, the Earth must be a sphere. Second, gnomons are very old, and Greek polymath Ptolemy had access to centuries of data tracking the angle and length of a gnomon’s shadow. His task was to find and count the intervals between the shortest shadow, giving him the dates of summer solstices.
Some years before Eratosthenes, Aristarchus had devised a methodology to determine the distance to the moon. As humans, we’ve been observing solar eclipses for millenia, so we knew they never last more than four hours. Since a solar eclipse is the moon traversing the Earth’s shadow which is roughly twice the Earth’s diameter, combined with the old knowledge that a lunar cycle is roughly 28 days, we can write an equality. We know that the ratio between 28 days (the lunar cycle) and four hours is roughly equivalent to the ratio between the distance to the moon and two Earth radii (alongside a constant factor of 𝜋). That’s as far as Aristarchus got, as he didn’t have any precise measurements for the key number: the Earth’s radius. Thanks to Eratosthenes, we can now determine the distance to the moon. What next?
For no reason at all, the apparent sizes of the moon and the sun (how big they are in the sky) happen to be identical. This miraculous coincidence unlocks the next rung of the ladder: the size and distance of the Sun.
Equipped with our knowledge of the Earth’s radius, the distance of the moon and its size, we can begin to think about determining the distance of the sun. Looking at the sky, the moon and the sun seem to be of the same size! This means that the ratio between the radius of the moon and its distance must be the same as the radius of the sun and the distance between the Earth and the sun. However, this only gives us a basic relation without any sense of scale, as the sun could be twice as far and twice as big as the moon while fulfilling our constraint. This is where the moon comes into the picture again: we know the moon has phases, and, after a little bit of thought, Aristarchus realized that when the moon is perfectly half-lit (first quarter), then the earth and sun make a right angle at the moon.
Since we know the distance between the earth and the moon, we only need to measure the angle between where in the moon’s orbit the half-moon occurs and the first quarter point in the moon’s orbit (that is, where the moon would be half-lit if the sun was infinitely far away). This is where the technology of the time hit a roadblock. Measuring that tiny angle was extraordinarily difficult and the best attempts of the time were off by a factor of 10. However, by this point, humankind had figured out how to climb the cosmic distance ladder and the path forward was clear. Just through human ingenuity, we had measured the circumference of the earth, the distance to the moon, and its size! The next rungs were only climbed around a thousand years later, bringing with them new brilliant methods. In a future edition, we shall explore them.