Updated January 4, 2024

*Image: abstract image representing the Pythagorean theorem AC ^{2}=AB^{2}+BC^{2} (Generated by an IA). *

We have no text from **Pythagoras** himself (≈570-495 BC).

All the writings that have come down to us on Pythagoras are of indirect origin. Most information about his works and ideas comes from later sources, mainly from authors such as Euclid (≈300 – ≈265 BC).

The **Pythagorean Theorem** first appears in a mathematical context in Euclid's "The Elements" (Book I, Proposition 47). “The Elements” are dated approximately to the 3rd century BC. BC, and it is possible that Euclid was not the sole author. This work is a compilation of thirteen books covering various aspects of mathematics and geometry. Although the original manuscripts have not survived to the present day, handwritten copies were made during Antiquity and the Middle Ages.

Thus, Euclid's writings provide one of the first known formulations of this theorem in a formal mathematical framework. Proposition 47 reads (modern translation):

“In a right triangle, the square constructed on the hypotenuse is equal to the sum of the squares constructed on the other two sides.”

Members of the Pythagorean sect were known for maintaining the confidentiality of their teachings and transmitted their knowledge orally. It is likely that the theorem was taught and used in this context.

It is possible that Pythagoras' disciples then wrote down his teachings, but no writing has survived time. Libraries and archives in antiquity were often destroyed by fires, wars or pillaging.

The writings of Greek historians and philosophers after Pythagoras are the main sources we have for knowing his life and thoughts.

The most important accounts of Pythagoras are those of later Greek historians and philosophers such as Diogenes Laertius (AD 180-240) and Iamblichus (AD 245-325).

Diogenes Laertius wrote "Lives and Doctrines of the Illustrious Philosophers", a biographical compilation of the lives of many ancient philosophers, including Pythagoras. Information about Pythagoras is based on various and sometimes legendary sources, and there is no guarantee that it is entirely true to the facts.

Iamblichus also wrote about Pythagoras in his work "Life of Pythagoras". However, as with Diogenes Laertius, it is essential to recognize that Iamblichus wrote several centuries after Pythagoras, and his work may reflect interpretations and theological elements of the time.

Pythagoras is often associated with the discovery of musical relationships and the influence of numbers on music. He studied the relationships between the lengths of vibrating strings and the frequencies of sound produced, thereby establishing connections between mathematics and music.

Pythagoras developed fundamental ideas in number theory. His fascination with numerical properties led him to explore the relationships between integers, particularly the properties of prime numbers and notions of perfect numbers.

Pythagoras discovered that the diagonal of a square whose sides measure 1 cannot be expressed as a simple fraction. This led to the notion of irrational numbers, which cannot be represented as ratios of integers.

Pythagoras founded the Pythagorean sect, a community where the study of mathematics was closely linked to philosophical, mystical and religious aspects. Members of the sect believed in the importance of numbers in understanding the world.

In Western Europe during the Middle Ages, medieval scholars and clerics generally did not have access to original Greek writings or direct Arabic translations.

The transmission of classical Greek knowledge occurred primarily through translations from Greek to Latin, carried out by medieval scholars. The Latin manuscripts containing these translations were often based on original Greek texts preserved in Byzantine libraries.

Although the original Greek texts have not survived, Greek copies and Latin translations were made in the Middle Ages, contributing to the preservation and transmission of their works in Western Europe.

However, some classical Greek texts were preserved in the Arab world, where they were translated into Arabic. Subsequently, some of these Arabic translations were rediscovered in Western Europe, and scholars undertook to translate these Arabic texts into Latin. Thus, although access to the original Greek writings was limited, there was indirect influence via Arabic translations.

Arab intellectuals fused Greek, Persian, Indian knowledge and other traditions with their own cultural heritage.

Arabs played a vital role in the preservation, transmission and translation of ancient knowledge during the medieval Islamic era (7th to 13th centuries).

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