Pythagoras’ Theorem

Foreword

We are all familiar with the famous Pythagoras’ Theorem. As we all know, the Pythagoras’ Theorem says that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. There have been more than 300 proofs to this theorem. Who actually discovered this theorem?

The Early Discovery

In around the 5th and 6th centuries BC, there was a “Pythagorean school” in Croton. Their motto was “all is numbers”. The school had significant contribution in areas like number theory, geometry, astronomy and music. The school had a strict rule which prohibited any publicity of its discoveries and creations to the outside world. There was a legend saying that after the school had discovered the Pythagoras’ Theorem, it celebrated by killing 100 cows. So the Pythagoras’ Theorem was also known as the “Hundred Cows Theorem”.

The Earliest Rigorous Proof

Since the Pythagorean school prohibited release of its discoveries to the outside world, there is no actual historical record about its discovery of the Pythagoras Theorem. The first one who gave a rigorous proof to the Pythagoras Theorem was the Greek mathematician Euclid. His proof given in Elements is the one that many modern mathematics textbooks adopt.

Contributions from Chinese and Egyptians

In the ancient Chinese text Zhoupi, one of the oldest mathematical and astronomical works in China, written in the 1st century AD, there recorded a conversation between the doctor Shang Gao in the Zhou Dynasty and Zhou Gong, . It was said that Xia Yu, at a time even earlier than the earliest records in Egypt, already knew to use the ratio 3 : 4 : 5 to form a triangle in the fight against flooding. Zhoupi even mentioned explicitly the method to compute the hypotenuse of a right triangle, namely, “to multiply each of the two legs by itself, sum them up and then take square root”. From this, we see that the Chinese at that time already knew the Kou-ku theorem (which is the Chinese way of calling the Pythagoras Theorem).

On the other hand, the mathematician Moritz Benedikt Cantor (1829 – 1920) believed that the ancient Greeks already knew about using right-angled triangles with side lengths in the ratio 3 : 4 : 5 to achieve purposes in surveying and architecture.

The Mystery of “Plimpton 322”

An ancient Babylonian tablet catalogued as “Plimpton 322” lists triples of positive integers like (3, 4, 5). The early scholars thought that it was just some accounting records in ancient times. The mystery was not solved until 1945 upon completion of the research taken by Otto Neugebauer and A. Sachs. It turned out that these triples are Pythagorean triples (a triple of positive integers which form the side lengths of a right-angled triangle is called a “Pythagorean triple”). The Pythagorean triples in “Plimpton 322” are very large. If the ancient Babylonians were not familiar with the Pythagoras Theorem and the parameters for Pythagorean triples, it would be impossible for them to construct such numbers. It is really amazing that the ancient Babylonians could make such outstanding achievements in around 2000 BC!