5 Mathematical Formulas from Ancient Times

It’s humbling to realize that much of high-school math, so vexing to so many of us, was already well understood thousands of years ago. The Egyptians came nowhere near E=mc2, but they knew how to find the volume of a pyramid. The Greeks didn’t conjure up calculus, but they did determine the area of a circle and proved it. Seen in historical context, these calculations are hardly less impressive than those of Einstein or Newton.

The modern world, with its digital computers and internal combustion engines, is built upon the exploration of numberphiles. Fortunately for us, they’ve been at it a long time.

“With the possible exception of astronomy, mathematics is the oldest and most continuously pursued of the exact sciences,” wrote the late David Burton, a professor at the University of New Hampshire, in The History of Mathematics.

Here are a few of antiquity’s greatest mathematical achievements, drawn from some of the earliest writings on the subject.

1. Volume of a Truncated Pyramid

Today the mind-blowing products of math are all around us, in every skyscraper and suspension bridge. Still, few inspire the same wonder as the Pyramid of Giza. Constructed some 4,600 years ago, it reveals the power of calculation like nothing else from the ancient world can.

As it happens, finding the volume of an entire pyramid is a cakewalk: take one-third of the area of the base, and multiply it by the height. It’s easy to figure this out with miniature models of a pyramid and a prism, both with the same base area and height. Fill the pyramid with water, pour it into the prism, and you find that the prism will hold exactly three times as much. Since the volume of a prism is simply base times height, you can use it to infer the volume of a pyramid.

The formula for the volume of a truncated pyramid (one with its top lopped off, also known as a frustum) is an order of magnitude more sophisticated. Just look:

The Egyptians didn’t write it like that, of course, but the Moscow Papyrus (a collection of math problems from roughly 1850 B.C.E.) shows that they grasped the underlying principle.

This was so far ahead of its time that Burton called it “the masterpiece of ancient geometry.” From a practical perspective, it allowed them to calculate, partway through construction, how much material they needed to finish the job.

Read More: Mathematical Solution Found For Jigsaw Problem We’ve All Faced

2. Pythagorean Theorem

If you remember anything from geometry, odds are it’s this: In a right-angled triangle, the square of the hypotenuse — the longest side — is equal to the sum of the squares of the other two sides.

You can verify this by literally drawing squares that jut out from each side. You can also express it as an elegant formula, which probably occupies about the same place in your memory as “the mitochondria is the powerhouse of the cell.”

Although this equation is named for the Greek mathematician Pythagoras, who lived in the sixth century B.C.E., it’s much older.

“We are often told that in mathematics all roads lead back to Greece,” Burton wrote. The Greeks themselves believed it had come down to them from Egypt, and modern archaeology largely supports that claim. But first dibs on this geometric gem goes to the Babylonians.

A clay tablet from roughly 1800 B.C.E., known as Plimpton 322, contains a list of Pythagorean triples — groups of three integers that satisfy the theorem. For example:

Many scholars have considered the list an exercise in pure, theoretical mathematics, perhaps even a problem set used for teaching students.

In 2021, however, Daniel Mansfield, a mathematician at the University of New South Wales, made the case for a more worldly purpose: This “study of rectangles,” he wrote, “seems to have originated from the problems faced by Mesopotamian surveyors measuring the ground.”

In a poem from the same period as Plimpton 322, one such surveyor reported that “when wronged men have a quarrel, I soothe their hearts.”

3. Quadratic Formula

The quadratic formula, another algebraic staple, is one of the first truly intimidating mathematical structures that high-school students encounter. Dig deep and you might even be able to summon the singsong tune your teacher used to lodge it in your brain. (Hint: “Pop Goes the Weasel.”)

Apologies for the retraumatization. To add insult to injury, Uta Merzbach and Carl Boyer wrote in A History of Mathematics that this infernal string of variables “afforded the Babylonians no serious difficulty.”

Perhaps it helped that they didn’t think of it in such an abstract sense. In fact, indefinite terms (like “a,” “b,” and “c”) had yet to be invented, so instead they used their words for “breadth,” “length,” “area,” and “volume.”

Like the Pythagorean theorem, the quadratic formula helped with on-the-ground administrative matters. But Merzbach and Boyer noted that many of the problems inscribed in Babylonian tablets “seem to be intellectual exercises, instead of treatises on surveying or bookkeeping, and they indicate an abstract interest in numerical relations.” Already people were beginning to see math as an end in itself.

Read More: Have You Heard The One About The Mathematician…

4. Thales Theorem

Time to give the Greeks their due. Among myriad geometrical innovations — looking at you, Euclid — they discovered this eminently cool fact: If you make a triangle using the diameter of a circle as one side, the other two sides (if they meet on the circle’s circumference) will always form a right angle. The proofs are too long to include, but you can see how it works here.

It may seem rudimentary now but remember that in the sixth century B.C.E., Thales and his contemporaries were inventing demonstrative mathematics from scratch — using logical reasoning to discover irrefutable truths about the world. He may well have learned the substance of his theorem during a trip to Babylon, but he was the one (at least according to tradition) who gave it an ironclad proof. That’s what makes it a “theorem,” not just a party trick.

“For this reason,” wrote Merzbach and Boyer, “Thales has frequently been hailed as the first true mathematician — as the originator of the deductive organization of geometry.”

There’s no way to be sure who really authored this theorem. Thales was a favorite, as was Pythagoras. But since the latter gets credit for the most famous equation of antiquity, we’ll let Thales have this one.

5. Archimedes’ Cattle Problem

A few centuries after Thales, the Greek mathematician par excellence was Archimedes. When he wasn’t busy revolutionizing geometry and inventing ingenious new tools, he sometimes amused himself with what Burton called “arithmetical problems clothed in poetic garb.”

He took inspiration for one such word problem from a line in the Odyssey: “Next you will reach the island of Thrinacia, where in great numbers feed many oxen and fat sheep of the Sun.”

Archimedes wanted to know just how many oxen there were. The question, as he posed it, was exceedingly complicated, but basically it came down to the difference between two squares, which can be represented thus:

Today this is known as a Pell equation, erroneously named for the English mathematician John Pell even though he was 2,000 years late to the game. That said, Archimedes and other early investigators of such equations were ill-equipped to solve them. Though he couldn’t have known it at the time, his original riddle asks for the solution to:

The answer — finally reached in 1965 with help from a computer — runs to 206,545 digits. As for Archimedes and his cohort, Burton wrote, “they probably displayed the equations involved and left the matter at that.” Yet that doesn’t detract from the imaginative force required just to frame the question.

Pell equations fascinated many of the modern era’s greatest mathematical minds, from Pierre de Fermat to Leonhard Euler. Before them, the Indian mathematicians Brahmagupta and Bhāskara II (living in the seventh and 12th centuries C.E., respectively) discovered algorithms for finding integer solutions to these equations. And even today people continue to sort out their subtler points, forming a scholarly throughline all the way 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.

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