The Mathematical Mystery Behind Mercator's Map

- Thank you to Squarespace for supporting PBS. In February, 1544, a man sits inside a cell inside a stone fortress in what is today Belgium, imprisoned by the Inquisition. Several of his fellow prisoners have already been burned at the stake, beheaded, buried alive. The charge against him is not for treason or rebellion. He's been charged with heresy for studying things that the church had decided were not meant to be studied. For asking forbidden questions about the world.

His name was Gerardus Mercator. And 25 years later, in spite of all of that, he published this. This is the most famous map in history. You've seen it in classrooms. It's a map most of us still use every day. And YouTube is full of videos explaining all the ways that this map is wrong, all the ways that it distorts reality. I mean, Greenland is big as Africa, like Europe, dominating the view, whatever's going on with Antarctica down here. What this was never meant to be the map that we use for everything.

It wasn't made to be hung in a classroom. It was built to solve one of the hardest problems of the age of exploration. And for that one thing, this map is a masterpiece. This is a video about something else because there's an even more fascinating mystery hiding in Mercator's map building. This map should have been mathematically impossible because it requires mathematics that didn't exist yet and that wouldn't be invented until half a century after Mercator published his map. So how did he do it and why?

This is the story of the best, worst map ever made. Hey, smart people. Joe, here. I wanna start with the problem that map was built to solve because it was a real problem and it was killing people in the 14 and 1500s. Explorers were sailing into uncharted ocean places that no European had ever seen, let alone mapped. And when you are in the middle of the Atlantic, no GPS, no radio, no satellites.

Figuring out where you are is sort of a matter of life and death. This is the most accurate map of the earth. I finally get to use that in a video. Problem is this is pretty useless for navigation. Let's say I wanna sail from Lisbon, Portugal to New York. If I wanna find the shortest route, I can do this. Notice that even though these two cities are basically the same latitude, the shortest path actually curves kind of north. That's because straight lines on a sphere are curves.

The shortest path between two points on the globe is curved, not straight. And we call these great circle roots because they always fall on circular paths positioned around Earth's center. These roots are easy to calculate and follow today with GPS and computers and everything. But to follow this curve route on a ship, you'd need to constantly recalculate your compass heading every few hours. And on a moving wooden ship in the middle of the Atlantic without a calculator and using 16th century math, that was basically impossible. Plus, do you know how much rum these guys were drinking? If only there was a way that sailors could just set their compass to one angle

and follow that bearing without touching it, it would be much easier and safer, even if it wasn't the shortest route on a globe. That path looks like this. Notice that every time it passes a vertical line of longitude, it crosses at the same angle. Set. Your cup is bearing once here and you get where you want to go. Easy peasy lemon squeezy. This kind of path is called a rum line, R-H-U-M-B. Not like rum like they were drinking on the ships all the time.

Rum lines actually follow spiral paths on a globe, not straight lines. It's also not the shortest path, which would be a great circle, but sailors were willing to make their journey a little bit longer if it meant not getting lost at sea. What sailors needed was a flat paper map where these spiral rum lines were straight and that they still met at each longitude line at the same angle. The man who solved this problem was named Gerardus Mercator. He was born Gerard Kremer in 1512, the seventh child of Cobbler in what's now Belgium.

His hometown sat just a few miles down river from Antwerp, then one of the wealthiest and most connected cities in the world. And most importantly, Antwerp in the early 15 hundreds was one of the busiest ports on earth. Hundreds of ships arrived and departed every single day loaded with goods and stories from distant lands. And we truly can't understand why Mercator created his map without thinking about the era that he grew up in. He was born just 20 years after Columbus arrived in America. Magellan circled the globe and Gerard was just around 10. As he was growing up, the world known to Europeans doubled in size and no one could keep track of it. For most of the thousand years

before Mercator, maps were not tools for getting from place to place. They were religious documents. They were made to tell a story about how the universe was ordered. This is one of the most famous maps of the medieval world. It was made around the year 1300. Look where Jerusalem sits exactly in the middle, geographically centered. Why? Because Ezekiel chapter five, verse five says, that's where it is: "Thus sayeth the Lord God; This is Jerusalem: I have said it in the midst of the nations and countries that are roundabout her." Who is gonna argue with that?

And on these maps, east was at the top because east was the direction of paradise. The Garden of Eden, this is literally where the word orientation comes from to orient oneself or your map meant to face east toward the orient toward paradise. And here's a map from the 12 hundreds. It showed the world as the literal body of Christ. Head up top in the east, hands on the sides, feet at the bottom, maps like these hung in churches across Europe. They were beautiful works of art, spiritually significant. But the point was not to draw the world as it actually is. Their point was to make scripture feel physically real. But good luck getting anywhere with a map like that.

So while the church is making maps like those sailors in the Mediterranean, were using something different. These are portal land charts. The first maps to draw coastlines accurate enough to navigate in real weather. And they weren't oriented toward the holy land. North is up top like we're used to. You see those lines running in every direction. Those are navigation lines that told sailors what compass bearing to follow to get from one port to another around the Mediterranean. And for the short distances of trade

around the Mediterranean, these maps worked fine. And these two types of maps coexisted for centuries without much problem because everyone understood that they each had their own special purpose. But then in 1406, everything changed. A Greek text by Claudius Ptolemy, written around 150 ad was translated into Latin for the first time. And this was shocking. Even though this was an ancient manuscript, it showed a world bigger than just about anyone knew existed. And it was charted with grids and coordinates.

It was based on mathematics, not faith. Gerard Kremer was born into a world struggling to reconcile all of this. The church's sacred geography, Ptolemy ordered mathematical view of the world and boats full of explorers that were coming home from places that before then no one in Europe even knew existed. The life of the man that we call Mercator was dedicated to figuring out one question, what did the world actually look like? Come with me, I need to show you something. And when Gerard enrolled in school as was standard practice, he translated his name to Latin Kremer, which meant "trader" or "merchant" became Mercator.

He studied under some of the greatest mathematicians in the Low Countries, learning how to make detailed astronomical instruments, learning spherical geometry and how to make globes. We're gonna get to the flat map, I promise, but it's worth taking a second to just nerd out on globes today. You can just order one of these on Amazon or whatever, right? But in the 15 hundreds globes were the ultimate status symbol. These were things that emperors and popes would have out to flex on common people to show off your knowledge and also let everybody know that you could afford one.

A globe was one of the most expensive things that you could own because making these was one of the most technically demanding crafts in Europe. It meant carving or turning or assembling a perfect sphere, hand engraving every detail in reverse on metal plates, and then printing these football shaped strips called gores and meticulously gluing them to the sphere with every seam perfect. And we think that Mercator ended up making hundreds of these globes during his life, but only a handful are still around today. And this is one of them.

Just think every time that he sliced the globe and tried to squash it flat, he had to wrestle with an almost impossible problem. How do you take a curved world and make it play nice on a flat page? And that question nagged at him for nearly 30 years until he came up with the map that changed history. But Mercator nearly didn't live to make his famous map. See, he was friends with some followers of one Martin Luther and the Catholic Spanish who ruled Mercator's homeland at the time.

They didn't much like that. And well, nobody expected what happened next. Nobody expects the Spanish Inquisition. In 1544, he was arrested for heresy. He spent seven months inside a fortress in the town that he was born. Four of his fellow prisoners were executed during that time. He survived apparently because some powerful patrons vouched for him, but he left his Catholic homeland and never went back. And in 1569, while he was in exile in Germany,

he published this the Mercator map. You remember those spiral rum lines? He had figured out how to turn rum lines into straight lines. It made long distance navigation with a compass, a cinch, and it spread across the world in the hands of sailors that used it. But there is a deeper mystery hidden in this map that took over 400 years to solve. And it has to do with how Mercator made it because it turns out that this map requires mathematics that didn't exist when he made it. Let's start with the fundamental problem.

This is more or less a sphere like the earth, but let's unpeel it. Well, part of our plan fell off. The table is flat, but no matter how hard we try, just can't get this orange peel to lie perfectly flat without tearing or wrinkling it. The fundamental problem is a geometric one. Now I'm gonna draw a line on this globe, not on my nice globe. I'm gonna do it on this cheap one. Let's start at the North Pole following this line of longitude down to the equator. Now let's take a 90 degree turn and draw a line along the equator until we reach this point.

Now let's take another 90 degree turn and head back up to the north pole. I've created a triangle. Look, there's three sides, three corners, but these angles are 90, 90 and 90. They add up to 270 degrees a triangle where all three corners are right angles on a flat surface that is impossible. The inner angles on two dimensional triangles always add up to exactly 180 degrees. A sphere and a flat plane are fundamentally different geometric objects. A mathematician named Carl Friedrich Gauss proved this in 1827.

You just cannot flatten a sphere without distorting it. So what do you do? You get stretchy. This came outta my closet. No comment. Let's see what happens when we take a two dimensional grid and try to stretch it over a three dimensional globe. It gets all distorted and weird doing the opposite, taking a spear and stretching it back to a flat surface. No matter what flat surface you pick, that distorts two, which means that every map is a lie. The only question is what kind of lie are you willing to tell?

I think we're done with this. What the heck? Cut. You can make a map that preserves area or one that preserves distances or one that preserves shapes and angles, but you can't do all three. It's like against the law mathematically. So Mercator made a choice. His map was designed to preserve angles and shapes that we see on the sphere in a rectangular map, even if that means that areas get totally screwed up, like Greenland looking as big as Africa. So here's how a globe becomes a Mercator map. These are those segments we were talking about earlier, Gores, that you make a globe out of. If I lay out these segments, you can see that we're left with these massive gaps near the poles, right?

We need to stretch these segments out to fill these gaps. So how do you do that? We have to force these curved edges into straight lines so that our lines of longitude run parallel. But up here farther from the equator, we have to stretch more than down here near the equator. At 60 degrees latitude, for example, we have to stretch by a factor of two and at the poles we have to stretch a finite point to infinity, which is why the poles aren't visible on a Mercator map. We can actually calculate how much we need to stretch at any point by looking at a globe. The equator is a circle with this circumference and any line of latitude.

Up here is just a circle with a smaller circumference. So any segment of latitude is just a segment of a smaller circle. Since the circumference of a circle simply scales with the radius, we can calculate how much smaller this circle of latitude is versus the equator, and that will give us the shrinking factor for any latitude above or below the equator. We want to calculate the smaller radius our prime. We know the larger radius are, and we know the angle here, theta, which is equal to our latitude.

The radius of our circle creates a right triangle, and in a right triangle, the ratio of our smaller radius to the hypotonus radius is equal to the co-sign of theta. This tells us how much any latitude is squeezed relative to the equator. But Mercator wanted to stretch those lines of latitude so that vertical lines of longitude were parallel. So the inverse of our squeezing factor tells us our stretching factor, the secant of theta. Now all lines of longitude are parallel on our map, but we've introduced a new kind of distortion by only stretching sideways.

Our map looks squished. The shapes of coastlines are all wonky. The solution Mercator applied the same stretching factor, the secret of theta vertically too. Whatever he stretched east west, he stretched by the same amount north south. This is why Greenland and Antarctica are so huge. They've been made wider and taller, but it's not a glitch. You see how those circles are still circles? That's the very thing that made this map so useful for navigation.

It preserves angles so far. That's math that most high schoolers would get today, and it was certainly known to people like Mercator. But here's where the map math mystery gets wild. To figure out exactly where to put each latitude line on the final map, you've gotta add up all of those stretch factors and not just the big ones. All of those little secant values for every infinitesimal step from the equator to the poles, every tiny step, you're applying that local stretch. You have to add them all up. The process of summing infinitely many infinitesimally small quantities that is called integration.

It's one of the foundations of calculus. And when you do it for the Mercator projection, the answer involves a special bit of math called a natural logarithm. But here's the thing, a mathematician named John Napier first developed logarithms in 1614. Isaac Newton and Gottfried Leibniz didn't develop calculus like integrals until the late 16 hundreds, but Mercator published his map in 1569. So Mercator made a map that requires math that wouldn't be invented until 45 years later. How the heck did he do it? Well, Mercator certainly never wrote it down. This was an unsolved mystery in the history of mathematics for hundreds of years. In 1599, like 30 years after Mercator first publishes his map, an English mathematician published a book

where he hand calculated how much to stretch latitude and longitude as you go away from the equator. It's basically a cheat sheet for making a Mercator map. By the 1640s, people were publishing all kinds of cheat sheets like this for multiplying and dividing large numbers. Let's say you want to multiply these two numbers. You can just open the base 10 logarithm table and look up this number. These are the exponents that we raise 10 to. In order to get those numbers, we just add these together.

Take that result, look it up in our logarithm table and see that it corresponds to this number logarithms. Were an easy way to multiply using simple addition. In the days before people had invented calculators and mathematicians realized that Mercator had basically hand calculated one of these logarithm tables decades before logarithms were invented to get those numbers. Mercator had basically made tiny infinitesimal slices up from the equator, smaller and smaller. It turns out early mathematicians like Archimedes did the same trick around 1800 years earlier. To calculate a value for pie,

Archimedes sliced a circle into thin triangle wedges. Then he compared the perimeter of this polygon inside the circle to one that touches the outside. Now keep dividing those into smaller wedges again and again, the more sides that you add, your shape gets infinitesimally close to a circle and you get a pretty darn close estimate for pie. It took until modern times for mathematicians to figure out that Mercator had basically done this same thing. Infinitesimally stretch out a sphere onto his flat map.

What he figured out by hand using ancient Greek geometry, was actually a whole new kind of math that hadn't even been invented yet during his life. Like I can barely solve a calculus problem with calculus, and this dude's living on hardcore mode. This is something that happens all the time in science and math. We don't talk about it enough. Mercator built a map that just worked by the time the map caught up to the map, sailors had already been using it for decades. But here is my favorite part of this story. That map from 1569, you probably used it today.

Open up your phone, open Google Maps or whatever your favorite map app is. Look at your neighborhood, your city block, a 90 degree turn looks like a 90 degree turn. The intersection down the street meets at the same angles it does in real life. Now, zoom out the Mercator projection. We use it every day because it does what we need a map to do every day at any scale, angles are preserved. That's what we expect when we navigate the world, right? For the world on here to look like the world out there, despite all the hate that the Mercator projection gets for that, it's the fricking best map out there.

Mercator spent his whole life chasing this world that was growing faster than he could map it, figuring out how to create a picture of the world on paper that was true to the world out there. He somehow did it before the math he needed was even invented. And what he built turned out to be wrong in exactly the right ways. So how did Mercator's map end up being the one that we use for basically everything, even though Mercator never meant it for that? Well, for starters, you can blame this guy. This is an actual Bible printed by Johannes Gutenberg, a German metalworker, an inventor of the movable type printing press.

Thanks to the printing press, one map could be copied and distributed really cheaply. These weren't valuable artifacts for rich people to flex anymore. It could be everyone's map. And that's exactly what happened. Like I remember one of these hanging in most classrooms when I was growing up, I'm pretty sure they weren't using it to teach us nautical navigation. But the fact remains, this map isn't an accurate representation of Earth. It makes places near the poles look way larger in area than they actually are.

Here's a website where you can go see that for yourself, like Greenland's big, but it's not that big. We can make this distortion visible. A French mathematician invented this method in 1859 on a globe. These circles are the same size and area on Mercator's map. They stay circles because angles are preserved, but they get huge near the poles area is not conserved. These circles let us see how different map projections distort the globe in their own way.

Here's another one. Areas are closer to reality, but things look weird because it's only stretched left to right. And here's another one developed by a German named Arno Peters in 1967, though it turned out it had already been invented by James Gall in 1855. So today we call it the Gall-Peters projection. This map was intended to be more politically correct by showing every landmass at its true relative size to others. Europe is way smaller than Africa because, well it is, but cartographers, they honestly kind of hated it because it cured one distortion by introducing new ones.

This and many other map projections were developed to give us different perspectives on the planet each for different reasons. But can some maps actually be harmful? So there's this idea that growing up staring at the Mercator projection gives us a biased sense of the world that the map that you grow up with shapes your view of who's big and who's small, which places matter more. But this actually doesn't seem to be the case. Researchers have tested this with tens of thousands of people asking them to compare two countries and guess their actual relative sizes, and people actually got it right most of the time.

Most of us don't really seem to over or underestimate sizes in our mental maps of the world the way that you might expect. Though there are some hints that if you grew up looking at the Robinson projection, you do slightly better than others. The most honest thing that you can do with any map is understand the specific ways that particular map is lying to you. But the fact of the matter is there's only one totally accurate view of the earth, and that's this. This map was built for ships, but sometimes when a tool gets used for something other than what it was designed for, it can help you solve problems you didn't even know you had.

It's true. This map is bad at some things, but not all of the things that you've heard. And it's amazing at many other things that we don't give it credit for. It's another reminder to. Stay curious. Hey, here's a fun fact for sticking around to the end of the video. Mercator is actually where we get the word atlas. That's the name that he gave to his collection of printed navigation maps. And we still use it today to describe a collection of maps. And speaking of mapping earth, happy Earth Month! PBS is celebrating Earth Month by releasing a ton of great new episodes across our channels,

diving deep into our amazing planet, like the latest episode of Overview, which is presented by a suspiciously handsome host. The episode explores the Berkeley Pit, one of America's most toxic Superfund sites that ironically may represent the future of environmental problem solving. There's links to that episode and the full Earth Month playlist down in description. And thank you to Squarespace for supporting PBS.

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helping us make videos like this one that are organic products of real human brains and take a lot of work. We'll see you in the next video, wherever you are on the map.