It’s relatively easy to measure the size of something small and nearby (just use a ruler), but it’s really hard to measure the size of something huge and far away. Just ask an astronomer (of which I am one), and they’ll agree (which I do). Because one of the big under-appreciated challenges in astronomy is figuring out how big things are and how far away they are. The history of astronomy and our understanding of the universe is intricately tied to the history of a bunch of clever humans coming up with ingenious ways to do just this.
And the cool thing for all of us math fans is that many of these methods are based upon very simple but also very powerful math—in particular, geometry. Today, I’d like to tell you a story of one of these methods that dates back more than 2,200 years. It’s the story of a brilliant Greek astronomer and mathematician named Eratosthenes and his efforts to measure the size of the Earth. How did he do it? Let's find out.
How to Measure Short Distances
Before we get to the cleverness that is Eratosthenes’ method for measuring the size of Earth, let’s take a brief minute to discuss the measurement of distances in general and to understand why the problem Eratosthenes set out to solve was such a difficult one.
As I said earlier, measuring the sizes of things and the distances to them is easy as long as they are small and nearby. If both of these things are true, you can just grab a measuring stick, walk over to the thing, and either measure its size or how far away it was from where you started. If the object or place you want to measure the distance to is too far away for your measuring stick, we have a few other options in the modern world. We can drive to the location and use the car’s odometer to tell us the distance, we can use some mapping software to calculate the distance, or we can use the GPS coordinates of two locations to figure out the distance between them.
Measuring the sizes of things and the distances to them is easy as long as they are small and nearby.
These things are all fairly easy to do, but some things aren’t so easy. How far away is the Moon? The Sun? The nearest star? Or some more distant star? As you can imagine, these are hard questions that require more than simply knowing how to read the different markings on a meter stick. To begin our exploration of the math behind these measurements, let’s now jump into our time machines and travel back a little more than 2,200 years to learn about how the ancient Greeks solved a problem that seemed unsolvable until a little geometry and a lot of human ingenuity was added to the mix.
Eratosthenes’ Big Idea
How big is the Earth? More specifically, what is the circumference of the Earth? That is the question that Eratosthenes was confronted with sometime around the year 240 B.C. By the way, if you think about it, you’ll realize that simply asking this question meant that—contrary to what many people believe—the ancient Greeks already knew that the Earth was round and not flat. And, as we’ll see, the very measurement we’re about to watch them make contains a form of geometric proof of the curved nature of the planet.
As the (true) story goes, Eratosthenes learned that at precisely noon on the first day of summer in the Egyptian city of Syene, a stick that had been placed in the ground vertically would cast no shadow. And if you looked down a very deep well at that moment in Syene, you could see the reflection of the Sun—something you didn’t see any other day of the year. Eratosthenes realized that this could only happen if the Sun was directly overhead at that moment so that the rays of light falling upon the Earth in Syene were precisely parallel to the stick and the well.
What really made this interesting to Eratosthenes was the fact that these same phenomena were not also visible on that day in his home city of Alexandria located roughly 800 kilometers (or 500 miles) north of Syene. Eratosthenes reasoned (correctly) that if the Earth were flat, and if the Sun were far enough away from the Earth so that its light rays all arrived at the Earth parallel to each another (which they more-or-less do), then every stick placed vertically in the ground at every position on Earth should cast the same length of shadow at the same time. The fact that sticks cast different shadows at the same time therefore serves as geometric proof that the surface of the Earth is curved and not flat.
How to Measure the Earth’s Circumference
But Eratosthenes didn’t stop there. He realized that he could use this observation and a bit of geometry to estimate the distance around Earth. Here’s the idea: As you travel away from Syene along the curved surface of the Earth, Eratosthenes reasoned that the shadow cast by a vertical stick at noon on the first day of summer should grow in length. And he further reasoned that if you measure the angle between the vertical stick and an imaginary line extending from the top of the stick to the end of the shadow on the ground, then you are actually measuring the angle between the two locations on Earth as seen from the center of the Earth.
To see that this must be true, take a look at the following picture and think about what would happen to the angles if the city of Alexandria is moved towards or away from Syene. As the angle between the cities increases, the angle made by the lengthening shadow in Alexandria also increases—and it grows by exactly the same amount as the angle between the cities.
Eratosthenes found that the angle to the end of the shadow in Alexandria was very close to 7 degrees. Since there are 360 degrees in a circle, he concluded that the city of Alexandria must be located about 7/360 or roughly 1/50 of the way around the entire Earth from Syene. He knew that the distance between the two cities was roughly 800 kilometers, so he calculated that the circumference of the Earth must be approximately 50 x 800 kilometers = 40,000 kilometers. Modern day measurements peg the size of the Earth at about 40,075 kilometers, which means that Eratosthenes was nearly spot on.
In truth, there were several assumptions about the geometry of the problem that he got slightly wrong, but they all ended up sort of cancelling each other out and resulted in an incredibly accurate calculation … especially considering it was made with very rudimentary equipment over 2,200 years ago. Which goes to show that a bit of math and a lot of ingenuity can take you a long way.
Wrap Up
Okay, that's all the math we have time for today.
For more fun with numbers and math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Public domain image: Eratosthenes' method for determining the size of the Earth
Tidak ada komentar:
Posting Komentar