Concepts explained
Pythagoras' Theorem — The Most-Used Formula There Is
Square two sides of a right triangle, add them, and you get the square of the third. At school it is a formula to memorise. In practice it reappears wherever land is measured, distance is computed, or the shape of space is questioned.
Five metres. You can know that without measuring.
Multiply 3 by itself and 4 by itself: 9 and 16. Add them: 25. Then find the number that gives 25 when multiplied by itself: 5. That is the whole thing.
Try other numbers. Six and eight: 36 plus 64 is 100, and the answer is 10. Five and twelve: 25 plus 144 is 169, and the answer is 13.
Such tidy combinations are rare. One and one gives 2, and the number that squares to 2 runs 1.41421356… without ever stopping. That number causes trouble later.
Why squares?
Why not just add the lengths? Three plus four is seven, but the rope is five. Not seven.
Because the relation is about areas, not lengths. Squaring a length means drawing a square on it. A three-metre side gives nine square metres, a four-metre side gives sixteen. Together, twenty-five — which is a square with a five-metre side.
A little further in
Egyptian surveyors knew that a rope knotted into twelve intervals, held at 3, 4 and 5, yields a right angle. After the Nile flooded, field boundaries had to be redrawn, and that required a reliable way to make a right angle.
Babylonian clay tablets carry the same relation. One tablet, thought to date from around 1800 BC, lists many combinations far larger than 3-4-5 — meaning they were not stumbled upon but generated by a method.
Using it and proving it
The Egyptian rope works at 3-4-5. It works at 5-12-13. But whether it works for every right triangle cannot be settled by checking examples, however many. You cannot measure all the triangles in the world.
What the Pythagorean school did was show why the relation must hold — by argument rather than measurement. This is where mathematics parts from surveying.
Euclid placed the theorem as the final proposition of Book I of the Elements, reachable only after the forty-six propositions before it. A whole book stacked up for one result.
The side effect that shook the school
Take a right triangle with both short sides equal to one. The square on the hypotenuse is two. So the hypotenuse is the number whose square is two.
The Pythagoreans held that everything in the world is expressible as a ratio of whole numbers — musical pitch, the motions of the stars. Yet the number falling out of their own theorem could not be written as any such ratio.
The story goes that the member who let this out was thrown into the sea. Whether it happened cannot be checked, but that such a story was told says something about what the discovery meant to them.
It becomes the definition of distance
Once Descartes gave every point in the plane a coordinate, finding the distance between two points became a right-triangle problem. Know how far across and how far up, and the hypotenuse is the distance.
From that moment the theorem stops being about shapes and becomes the definition of distance. It extends the same way into three dimensions, and into four or more. It is still the formula a computer uses to measure between two points.
Where the formula gives way
Riemann took one more step. Measuring distance this way assumes space is flat. If space curves, each point needs its own rule.
A globe makes it easy to see. Two people setting off north from the equator start out parallel and meet at the pole. Pythagoras' formula does not hold there as written.
distance² = across² + up². The same rule everywhere.
Each point has its own rule. How much it differs is how much it curves.
General relativity did not discard the theorem; it extended it. Where there is no gravity, the original formula returns. A formula that began with one triangle 2,500 years ago ended up writing down the shape of the universe.
The question that remainsThe Egyptians could use the relation; the Pythagoreans proved why it holds. What lies between knowing how to use something and knowing why it works?