It’s humbling to know that much of the high school mathematics that so bothers so many of us was already well understood thousands of years ago. The Egyptians were nowhere near E=mc2but they knew how to find the volume of the pyramid. The Greeks did not come up with calculus, but they determined the area of a circle and proved it. In historical context, these calculations compare favorably with those of Einstein and Newton.
The modern world, with its digital computers and internal combustion engines, is built on the exploration of numbers enthusiasts. Luckily for us, they’ve been at it for a long time.
“Aside from astronomy, mathematics is the oldest and most continuously pursued of the exact sciences,” wrote the late University of New Hampshire professor David Barton. history of mathematics.
Here are some of the great mathematical achievements of antiquity, excerpted from early writings on the subject.
1. Volume of truncated pyramid
Today, the wonders of mathematics are all around us, in every skyscraper and suspension bridge. Still, few evoke as much wonder as the Pyramids of Giza. Built around 4,600 years ago, it reveals a power of calculation like nothing else in the ancient world.
As it happens, finding the volume of the entire pyramid is easy. Just take one-third of the base area and multiply it by the height. This is easily understood using miniature models of pyramids and prisms, which have the same base area and height. If you fill the pyramid with water and pour it into the prism, you will find that the prism can hold exactly three times as much. Since the volume of a prism is simply the product of its base and height, we can use this to infer the volume of a pyramid.
The formula is volume of truncated pyramid (truncated at the top, also called frustum) is an order of magnitude more sophisticated. Take a look:
truncated pyramid type
Of course, the Egyptians didn’t write it that way, but the Moscow Papyrus, a collection of math problems from about 1850 BC, shows that they understood the underlying principles. .
It was far ahead of its time, and Burton called it “a masterpiece of ancient geometry.” From a practical point of view, halfway through construction, it became possible to calculate the amount of materials needed to complete the job.
read more: A mathematical solution to the jigsaw problem that everyone has faced is found
2. Pythagorean theorem
If you remember anything about geometry, it’s probably this. In a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
You can see this by literally drawing a square sticking out from each side. It can also be expressed as elegant formulaThis probably occupies about the same place in your memory as “mitochondria are the powerhouse of the cell.”
This equation is named after the Greek mathematician Pythagoras, who lived in the 6th century BC, but it is much older.
“In mathematics, it is often said that all roads lead to Greece,” Burton writes. The Greeks themselves believed it came from Egypt, and modern archeology largely supports that claim. However, the Babylonians were the first to understand this geometric gem.
Clay tablet from about 1800 BC known as Plimpton 322contains a list of Pythagorean triples (groups of three integers that satisfy the theorem). for example:
Many scholars consider this list to be a purely theoretical mathematics exercise, or even a problem set used to teach students.
However, in 2021, mathematician Daniel Mansfield from the University of New South Wales argued that: more worldly purposes: This “study of rectangles,” he writes, “seems to have originated from problems faced by Mesopotamian surveyors measuring the ground.”
In a poem from the same era as Plimpton 322, one such surveyor reports, “When the wronged men quarrel, I soothe their hearts.”
3. Quadratic formula
of quadratic formulais a staple of algebra and one of the first truly scary mathematical structures that high school students encounter. If you dig deeper, you may even be able to recall the tune of a song that your teacher has imprinted in your brain. (Hint: “Pop goes Weasel”)
I apologize for the re-traumatization. To add insult to injury, Uta Merzbach and Karl Boyer wrote: history of mathematics This hellish array of variables “posed no significant hardship to the Babylonians.”
Perhaps it helped that they didn’t think of it in such abstract terms. In fact, indefinite terms (such as “a,” “b,” and “c”) had not yet been invented, so words for “width,” “length,” “area,” and “volume” were used instead. It had been.
Like the Pythagorean theorem, the quadratic formula was useful for management problems in the field. However, Merzbach and Boyer note that many of the problems inscribed on the Babylonian tablets “appear to be intellectual exercises rather than treatises on surveying or bookkeeping, indicating an abstract interest in numerical relationships. ” he pointed out. People were already beginning to think of mathematics as an end in itself.
read more: Have you ever heard a story about a mathematician…
4. Thales theorem
The time has come to give the Greeks their due. Among their countless geometrical innovations, Euclid, they discovered this most wonderful fact. If you create a triangle using the diameter of a circle as one side, the other two sides (if they intersect on the circumference) always form a triangle. Right angle. The proof is too long to include, but you can see how it works here.
It may seem rudimentary now, but in the 6th century BC, Thales and his contemporaries developed empirical mathematics, the use of logical reasoning to discover irrefutable truths about the world. Remember, we were inventing from scratch. It is quite possible that he learned the essence of his theorem during his trip to Babylon, but it was he who (at least according to tradition) provided an ironclad proof of it. That’s why it’s a “theorem” and not just a party trick.
“For this reason, Thales has often been hailed as the first true mathematician, the originator of the deductive organization of geometry,” Merzbach and Boyer write.
There is no way to be sure who actually created this theorem. Thales was a favorite, as was Pythagoras. But the latter is credited with the most famous equation of antiquity, so let’s give this one to Thales.
5. Archimedes’ Cow Problem
Centuries after Thales, the greatest Greek mathematician was Archimedes. When he was not busy revolutionizing geometry, invent creative new toolsHe sometimes enjoyed what Burton called “arithmetic problems in poetic costume.”
He drew inspiration for one such word problem from a passage. odyssey: “Next you arrive at the island of Thrinakia, where many cows and fat sheep of the sun are feeding.”
Archimedes wanted to know how many cows there were. The problem he posed was very complex, but it basically comes down to the difference between two squares. This can be expressed as:
Archimedes’ cow problem
Today, this is known as the Pell equation, which was mistakenly named after the British mathematician John Pell, even though he was 2,000 years late. However, Archimedes and other early researchers of such equations did not have the ability to solve them. Although he could not know it at the time, his first riddle called for a solution:
Archimedes’ cow problem, b
The answer was finally arrived at with the help of computers in 1965 and had 206,545 digits. As for Archimedes and his friends, Burton wrote, “They probably displayed the relevant equations and left the problem alone.” But that doesn’t take away from the imagination needed to formulate questions.
Pell’s equation has fascinated many of the great mathematical minds of our time, from Pierre de Fermat to Leonhard Euler. Before them, the Indian mathematicians Brahmagupta and Bhaskara II (who lived in the 7th and 12th centuries CE, respectively) discovered algorithms to find integer solutions to these equations. And even today people continue sort out their subtletiesforming a scholarly consistency dating back to ancient times.
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Cody Cottier is a contributing writer at Discover who loves exploring big questions about the universe and our home planet, the nature of consciousness, the ethical implications of science, and more. He holds a bachelor’s degree in journalism and media production from Washington State University.
