## Columbus did not risk falling over the edge of the world, but his first voyage was far more risky. In fact, no one held the world to be flat. Those who dared oppose Columbus relied on a librarian who measured the world with a stick.

**Pop quiz: for long have we known that the Earth is not flat? Or in more general terms: for how long have we as a species pondered the physical shape of the ground beneath us? And in epistemological terms: when did we realize that the shape of the physical world is even a valid question?**

### Ships falling off the Earth

**My geometry students answered the Flat Earth question as I thought they would: **Columbus, 1492, three ships and quasi-Indians. He sails west, doesn’t fall off the edge of the Earth. According to legend, Columbus set out to find a shorter route to Asia by heading west and swinging round the backside of the planet. Led by intuition to believe that the Earth is spherical, our brave captain countered the belief that he would drop off the side of, well, everything. His critics warned him off, queen Isabella sponsored him, and he heroically set foot on the New World, to the score of an awesome soundtrack. Fair enough, but we need to correct one small detail here: Columbus was wrong; his critics were right.

**There really was no controversy regarding whether Earth has a backside to traverse,** but rather her size. Columbus did not stand up against Flat Earthers. He believed the Earth to be handily small, while his opponents warned that he and his crew would never last the long voyage — excellent advice. To project a Flat Earth belief on 15^{th} century scholars is but an arrogant misconception of a disneyesque Dark Age.

**Literature has undeservedly ascribed the Catholic church with a ** belief in a Flat Earth and a suppression of rational thought. Not so; for example Saint Augustine merely that the Southern Hemisphere was uninhabited. The equator was held as a scorching desert, impossible to cross, and since

- all humankind is uniquely descended from Adam
- the Equator cannot be crossed
- humankind is present on the northern hemisphere

it follows that mankind was spawned in the north exclusively, unable to visit South Side, and thus the Southern Hemisphere is uninteresting for trade and taxes. (though modern readings of Augustine suggest that he did describe the Earth as being “the bottom of the universe”, layered beneath water, air and fire. The Columbus controversy regarded the circumference of the Earth, which implies that not only was there a rough figure for the size of the planet, but several conflicting ones. The mainstream, non-Christopher measurement was even quite good. Now, the question: how to measure a planet?

### Measuring Earth’s Circumference

**Enter, stage left, Erastothenes of Alexandria, librarian.** In his work, he came across a passing reference to a city in Egypt — Cyene, modern-day Aswan — where on a certain day of the year, the sun shines straight into the bottom of a well. Being a rather sharp fellow, he knew that meant the sun was at zenith at that particular day. This enabled him to perform an amazing geometrical stunt: estimating the size of the Earth.

**Turning a crank half a turn makes the handle travel half the distance around the full circle.** A quarter turn makes the crank travel a quarter of the distance. One seventh of a turn makes it travel one seventh of the distance. So, there is a simple relation between a partial angle and a partial arc length: if you turn the crank a part of the turn, the handle travels just as big a part of the distance. This is true for any circle, for instance the one described by the Earth’s circumference.

**There is an angle between the two cities from the Earth’s core, and there is a travel distance between them** measured from Aswan to Alexandria. The distance is known, but the angle is not. Looking at the Earth as a whole, the situation is the opposite: we do not know the distance around the entire planet, but we know the angle: 360º — just as with any circle. This is an amazing juxtaposition: we know the “local” distance, but not the “local” angle. We know the “global” angle, but not the “global” distance — just as with the crank handle, or any circle.

**If we know how the local angle compares to the global angle** (360º), we know how the local distance compares to the global distance, and vice versa. For Erastothenes: if he could figure out the angle between the cities and how it compares to 360º, he could also figure out how the distance between the cities compares to the circumference of the Earth. If the angle between the cities turned out to be, say, 1/8^{th} of a full circle, then the circumference of the Earth must be eight times longer than the ground distance between the cities. He did figure out that city-separating angle, and according to legend, he did it with a wooden stick.

Let’s start with a couple of assumptions:

**Sunrays are parallell.**This is not absolutely true, but “true enough”. The sun is at a tremendous distance, and the difference in sizes obliterates any angles.**Aswan lies roughly 800 kilometers due south of Alexandria.**This estimate was sourced from many, many camel-driven trade caravans, numerous enough to average a good guess. Thus, we have a good value for the local distance.

### Erastothenes’ Experiment

**On June 21 ^{st}, the Sun’s rays shone straight into the well in Aswan** (the sun hung directly over the city). They also shone upon the city of Alexandria, but since Alexandria lies further north, they entered at an angle (the sun appeared lower in the sky than in Aswan). A pole driven into the ground at Aswan would not cast a shadow, but it did in Alexandria. By measuring the length of the pole and the shadow, it is easy to calculate the Sun’s elevation above the horizon. In Erastothene’s case, he found that the sun hung roughly 83 degrees above the horizon, making it easy to calculate . Let’s take a look at the situation now.

**We are still interested in the separation between the cities. **Instead of calculating the angle of elevation of the Sun, we can calculate the angle marked in the diagram — the angle “under the pole”. It turns out that this angle is exactly the same as the city-separating angle! This is dependent on the assumption that the sun rays are parallel, which, as noted above, is “true enough”. A slanted line crossing two parallel lines will produce two identical angles at each intersection. Try it: tilting the crossing line (the “transversal”) will make one angle narrower and widen the other by the same amount. The exact opposite will happen at the other intersection, keeping the angles identical regardless of the angle.

**This is the exact same system as Erastothene’s set-up:** the Sun’s rays form the two parallel lines, and the line from the Earth’s core to Alexandria produces the transversal. So, by measuring the angle at the top of the pole — a very mundane task — we find the city-separating angle at the center of the planet.

**Then the deal is settled:** we know the local angle between the cities, the local distance between the cities, and the global angle (the full 360º), missing only the global distance (the circumference of the Earth). The distance between Aswan and Alexandria was then taken to be 5000 stadia (approximately 800 kilometers), the angle of separation (again, given by the Sun’s elevation in the sky at Alexandria) turned out to be a little more than 7º, and the global angle is ever 360º.

### Using the Wrong Measure

**The only stone left unturned is the “stadia” measure mentioned above,** the unit Erastothenes gave his result as: the circumference of the world turned out to be 252000 stadia. The stadion is an ancient measure of distance, and in itself not problematic. But, there were two systems in use in Erastothenes’ time: the Attic (greek) and the Egyptian, and no-one knows which he used. If it was the Attic one, his calculation was off by just about 15%, but if he used the Egyptian stadion, his value would imply that a trip around the Earth was 39690km, which is off by less than one percent.

**Regardless of accuracy, his triumph was to devise a method** of calculating the circumference, where earlier no-one knew anything about its size. Then, using this value, Erastothnes leap-frogged through the solar system: he calculated the size and distance of the moon and the Sun, but that’s another story. The circumference of the Earth is almost exactly four million meters, but the reason for that is also another story.

**We return to Christopher Columbus.** He set out westward to find a shorter route to Asia, specifically to make the jump from the Canary Islands to Japan, but he was wrong regarding the distance. Ironically, his distance estimate was correct, but in the wrong unit: he mistook the longer Arabic mile (1.8km) with the Italian mile (1.2km). All in all, he estimated the trip between the Canaries and Japan to be 3700km, while it in practice is 19600km. This is the controversy he fought, not that he would fall off the Earth. Sailors and scholars alike knew that he’d never last the trip, but he persisted and sailed off into the sunset. Luckily, Chris bumped into, uh, the Americas.

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