From Babylon to Modern Times: The Thousand-Year Journey of a Theorem
A journey through the history, geometry, and magic of right triangles
☕️ Welcome (Back) to the Math Café
Dear math friend,
I’m so glad to see you here!
For this Café edition, we’ll travel together through a story that spans thousands of years —
The story of a theorem you recently proved in our puzzles: the Pythagorean Theorem!
🏺 The Story: From Ancient Tablets to Greek Temples
You probably learned the theorem back in junior high school and didn’t think much about it:
In a right triangle, the square of the largest side — the hypotenuse, the side opposite the right angle — is equal to the sum of the squares of the two other sides.
You may even have learned a proof… and soon forgotten it (I know I did 😅).
But this familiar statement actually played an extraordinary role in the history of mathematics!
The first traces of the theorem go back more than 4,000 years.
A clay tablet from the First Babylonian Empire (c. 1900 B.C.E.), now known as Plimpton 322 and kept at Columbia University, lists columns of numbers that we now call Pythagorean triples: sets of integers (a, b, c) satisfying
a²+b²=c².
This strongly suggests that Babylonian scholars already knew at least some version of the theorem — not surprising given its role in astronomy, and the Babylonians’ advanced knowledge in the field.
In ancient India, the Baudhāyana Śulba-sūtra (circa 800–400 B.C.E.) states the theorem explicitly in special cases.
Meanwhile, ancient Egyptians regarded the 3−4−5 triangle as sacred: the hypotenuse (of length 5) represented the god Horus, and the other two sides (of lengths 3 and 4) gods Isis and Osiris (as described here by Greek historian Plutarch).
Centuries later, the theorem became central to the mystical school founded by the Greek philosopher Pythagoras (6th century b.c).
Legend has it that, upon proving the result, Pythagoras sacrificed a hundred oxen to the gods in gratitude, though modern historians suspect he may have learned it during his travels (for example in Egypt) and shared it with his Greek contemporaries, who were unaware of the earlier discoveries.
Nowadays, far from the mysticism of olden time, the theorem is a standard component of most junior high schools/ high schools curriculums around the world — with around 50 million students learning the theorem each year! Its generalizations play an important role in many fields, including civil and structural engineering, game design (normalizing movement speed, projectile aiming, collision detection, physics and camera calculations), GPS & satellites positioning (using spherical analogues of the theorem to determine distance formulas), and medical imaging (reconstructing 3D structures from 2D using MRI, CT or ultrasound).
✏️ Your Turn!
Now it’s time to challenge yourself!
Below are two ancient diagrams. Each one hides a proof of a result related to the Pythagorean theorem. Can you spot it?
(—> Scroll down for the solutions!)
1️⃣ Zhoubi Suanjing’s diagram (China, ~100 B.C.E.)
Puzzle difficulty: ☕️☕️☕️
Hint: This diagram hides a geometric proof that the hypothenuse of a right triangle with sides’ length 3 and 4 has length 5.
2️⃣ Diagram by Sir Thomas Heath (20th century)
Puzzle difficulty: ☕️
Hint: This modern reconstruction illustrates a purely visual proof of the theorem itself.
🌱 Next Week
Next week, you’ll receive a freshly brewed selection of new puzzles — designed to both stimulate and delight your mathematical curiosity.
Until then, I wish you a beautiful and meaningful week!
With love and curiosity,
Sybille ☕️
✏️ …. Solutions ….
1️⃣ The Zhoubi Suanjing’s 3-4-5 Proof (c. 100 B.C.E.)
2️⃣ Rearrangement Proof of Pythagoras Theorem
Rearrangement proof attributed to Sir Thomas Heath (20th century).
