The Pythagorean Theorem Makes Construction and GPS Possible
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OK, time for a pop quiz. You’ve got a right-angled triangle — that is, one where two of the sides come together to form a 90-degree angle. You know the length of those two sides. How do you figure out the length of the remaining side?
That’s easy, provided that you took geometry in high school and know the Pythagorean theorem, a mathematical statement that’s thousands of years old.
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The Pythagorean theorem states that with a right-angled triangle, the sum of the squares of the two sides that form the right angle is equal to the square of the third, longer side, which is called the hypotenuse. As a result, you can determine the length of the hypotenuse with the equation a2 + b2 = c2, in which a and b represent the two sides of the right angle and c is the long side.
A pretty slick trick, huh? But the man whom this math trick is named for is nearly as fascinating. Pythagoras, an ancient Greek thinker who was born on the island of Samos and lived from 570 to 490 B.C.E, was kind of a trippy character — equal parts philosopher, mathematician and mystical cult leader. In his lifetime, Pythagoras wasn’t known as much for solving for the length of the hypotenuse as he was for his belief in reincarnation and adherence to an ascetic lifestyle that emphasized a strict vegetarian diet, adherence to religious rituals and plenty of self-discipline that he taught to his followers.
Pythagoras biographer Christoph Riedweg describes him as a tall, handsome and charismatic figure, whose aura was enhanced by his eccentric attire — a white robe, trousers and a golden wreath on his head. Odd rumors swirled around him — that he could perform miracles, that he had a golden artificial leg concealed beneath his clothes and that he possessed the power to be in two places at one time.
Pythagoras founded a school near what is now the port city of Crotone in southern Italy, which was named the Semicircle of Pythagoras. Followers, who were sworn to a code of secrecy, learned to contemplate numbers in a fashion similar to the Jewish mysticism of Kaballah. In Pythagoras’ philosophy, each number had a divine meaning, and their combination revealed a greater truth.
With a hyperbolic reputation like that, it’s little wonder that Pythagoras was credited with devising one of the most famous theorems of all time, even though he wasn’t actually the first to come up with the concept. Chinese and Babylonian mathematicians beat him to it by a millennium.
“What we have is evidence they knew the Pythagorean relationship through specific examples,” writes G. Donald Allen, a math professor and director of the Center for Technology-Mediated Instruction in Mathematics at Texas A&M University, in an email. “An entire Babylonian tablet was found that shows various triples of numbers that meet the condition: a2 + b2 = c2.”
The Pythagorean theorem isn’t just an intriguing mathematical exercise. It’s utilized in a wide range of fields, from construction and manufacturing to navigation.
As Allen explains, one of the classic uses of the Pythagorean theorem is in laying the foundations of buildings. “You see, to make a rectangular foundation for, say, a temple, you need to make right angles. But how can you do that? By eyeballing it? This wouldn’t work for a large structure. But, when you have the length and width, you can use the Pythagorean theorem to make a precise right angle to any precision.”
Beyond that, “This theorem and those related to it have given us our entire system of measurement,” Allen says. “It allows pilots to navigate in windy skies, and ships to set their course. All GPS measurements are possible because of this theorem.”
In navigation, the Pythagorean theorem provides a ship’s navigator with a way of calculating the distance to a point in the ocean that’s, say, 300 miles north and 400 miles west (480 kilometers north and 640 kilometers west). It’s also useful to cartographers, who use it to calculate the steepness of hills and mountains.
“This theorem is important in all of geometry, including solid geometry,” Allen continues. “It is also foundational in other branches of mathematics, much of physics, geology, all of mechanical and aeronautical engineering. Carpenters use it and so do machinists. When you have angles, and you need measurements, you need this theorem.”
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The Pythagorean Theorem Makes Construction and GPS Possible
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