You might be a flat-earther if...

I see a lot of flat-earth stuff on Facebook. I don't know how if any of it is sincere, or if most or all of it is just engagement farming or flamebait. I just know I can't look away. The amount of utter ignorance apparently on display astounds me.

It's been well understood for millennia that the earth was spherical. Simple observation proved that: ships leaving port didn't just appear to get smaller as they sailed away; they disappeared from the bottom up as though going over a hill. The shadow cast on the moon during a lunar eclipse was always a circle; only a sphere casts a circular shadow from all angles.

In the third century BC, Eratosthenes measured the circumference of the earth using a couple of sticks and high-school geometry. Some of his assumptions were off (for example, the exact distance and bearing from Alexandria and Syene), but his methodology was sound. Columbus wasn't trying to prove the world was round; that was understood. The dispute was over its size, and whether Asia wa reachable by sailing west from Europe. (Obviously, it wasn't. It was fortunate for Columbus that North America was in the way. He thought the globe was much smaller.) Enlightenment intellectuals in the 18th and 19th centuries concocted the myth that the ancients believed the world to be flat, in order to portray religion as anti-reason. In reality, educated people knew the earth was a sphere. That people today will seriously entertain the notion that we live on a planar surface just goes to show how un-educated we have become.

Here are two basic proofs that the earth is spherical—one negative and one positive—that should be understandable to anyone. I'm no scientist; my science and math education is high-school level (with a year of university engineering thrown in; I dropped out). Fortunately, all you really need is a good grasp of basic geometry.

A negative proof

Fundamentally, flat-earth geography simply does not work. The majority of flat-earthers will point to an azimuthal equidistant projection such as Gleason's New Standard Map of the World, which is consistent with flat-earth claims of the earth being a disc with the North Pole at the centre and an "ice wall" around its circumference. An azimuthal projection is useful for representing accurate distances from the centre point. However, on Gleason's map, the farther south you get, the more distorted the projection gets. Southern continents like Australia get stretched in the east-west direction, and Antarctica, which is larger only than Australia if you exclude its ice shelves, becomes a massive ring around the perimeter, bigger than all the other land masses together. Two southern-hemisphere cities like Sydney, Australia and Buenos Aires, Argentina are approximately 11,800 kilometers apart, but appear practically on opposite sides of the map. Tokyo is about 18,400 km from Buenos Aires, but it appears closer than Sydney.

Different map projections are useful for certain purposes by preserving some characteristic of the Earth's geography, but no flat map can perfectly depict the geography of Earth in every respect: each one still distorts distance, bearing, direction, shape, and/or area. That's because every map is a projection of a curved surface. Only one map can accurately represent all of those characteristics. That is a globe.

Flat-earthers don't even agree on the map. Most advocate for an azimuthal projection, but some will dispute whether the earth has one pole or two, whether Antarctica is a regular continent or an "ice wall" ringing the world, and so forth. With the surveying technology available to us today, you might imagine that producing an accurate flat map of a flat surface would be easy. But it's never been done—or, as far as I know, even tried. Flat-earthism utterly fails to describe the world as we know it to be.

A positive proof

Sailors figured out very early on how to determine their latitude by measuring the altitude of certain celestial bodies, such as the sun or Polaris, the North Star. We all know that Polaris sits almost directly above the North Pole, making it appear as though the entire (northern) sky rotates around it. By sighting Polaris, measuring its altitude (angle above the horizon), a navigator could directly measure his latitude. For example, I live in Ottawa, which is at 45.4° north. If I look at Polaris at night, my sight line is at about 45 degrees to the horizon.

Consider the following locations in the Americas:

• Resolute, Nunavut
• Whitehorse, Yukon
• Ottawa, Ontario
• New Orleans, Louisiana
• Guatemala City, Guatemala
• Quito, Ecuador

I've chosen them because their latitudes are approximately 15-degree intervals. If they were aligned on the same line of longitude, they would be pretty evenly distributed at around 1,700 km apart.

Sightings of Polaris from a globe

To a person at the North Pole, Polaris is directly overhead: an altitude of 90°. In Resolute, he would see it at an altitude of 75°; in Whitehorse, at 60° and so forth. Quito is almost directly on the equator, so Polaris would have an altitude of 0°—directly on the horizon.¹ If you could draw those sight lines in the air from the observers to Polaris, you would discover that they are, for all practical purposes, parallel (Polaris is 450 light years from earth, so the angle of the light that reaches the North Pole and Quito differs by trillionths of a degree). Clearly our observers are all looking at the same thing.

Sightings of Polaris from a flat earth.

But if we assume the surface of the earth is flat, we have a problem. In Ottawa² I sight Polaris at a 45° altitude. But someone in New Orleans sighting at an altitude of 30° is looking right past Polaris. In fact, it's evident that these seven observers are either not looking at the same object, or they're not observing it at the correct angle, or we're mistaken about their latitude. They couldn't actually be evenly spaced; the distance between them would be greater the farther south we went—and division by zero would put Quito an infinite distance away from the North Pole!

Even more, if the altitude of Polaris in Ottawa is 45°, then Polaris, the North Pole, and Ottawa are the corners of a right isosceles triangle. That means Polaris is as far above the North Pole as the North Pole is from Ottawa: 5,000 km. But if you're viewing Polaris from New Orleans (6,675 km from the North Pole) at an angle of 30°, then it must be only 3,854 km above the pole.

Celestial navigation is based on the earth having a spherical surface; indeed, it only works on a globe. Try to use the same methodology to navigate on a flat earth, and the math is contradictory and incoherent. You'll get lost.

You don't need to be an astronomer or scientist to understand why the earth is not flat. I'm not. I just paid attention in class. How can you tell if someone didn't pay attention? They submit obvious howlers as devastating proof that the earth is flat.

• If you believe the earth's surface is planar, but you can't find an accurate map of it, you might be a flat-earther.
• If you believe the sun and moon are small and located just above the earth, but you can't explain why they're not always visible everywhere all the time, you might be a flat-earther.
• If you believe Antarctica is actually an "ice wall" surrounding the earth, but you can't explain how a small and local sun can provide 24-hour daylight in the south from September to March while leaving the North Pole in 24-hour darkness, you might be a flat-earther.
• If you believe there is no such thing as the south pole, but you can't explain why the sky appears to rotate around two points—one in the north and one in the south—you might be a flat-earther.
• If you can't explain why the sky rotates clockwise in some places and counterclockwise in others, you might be a flat-earther.
• If you can't explain why someone in Australia can never see the North Star or someone in Canada can never see the Southern Cross, you might be a flat-earther.
• If you are an Australian, you might inexplicably be a flat-earther.
• If you believe that an azimuthal equidistant projection map is the best representation of the earth, but you can't explain why the shortest distance for direct flights between Australia and Chile don't go over the Arctic, you might be a flat-earther.
• If you claim that on a globe you shouldn't be able to see a city skyline from 100 km away because of the curvature, but you can't explain why you don't see the bottom half of the buildings, you might be a flat-earther.
• If you can't explain why there's a horizon at all, you might be a flat-earther.
• If you say gravity is really just density or buoyancy, but you can't identify the force causing denser or less buoyant objects to fall downward instead of some other direction, you might be a flat-earther.
• If you fail to understand that you can't explain density or buoyancy without gravity, you might be a flat-earther.

Footnotes

1. To be precise, Quito is just south of the equator—just far enough that you can't actually see the North Star there. For the sake of argument, I'm taking a little creative liberty and assuming it sits directly on the equator.
2. I'm using Ottawa as my reference simply because that's where I am and what I can observe. The argument here would be the same anywhere else.