Euclid, lived in AD **Alexandria, Egypt **around 300 BC, was a mathematician and philosopher, best known for writing the **treaty *** Elements *and in particular for the two theorems that bear his name, which we all studied at school. He also dealt with astronomy, music and mechanics. Very little is known about his life, but it is certain that he was a teacher at the

**museum**, the main scientific institution of the time. His most famous work, the

*Elements*collects and systematizes the mathematical thought of the ancient world, inventing a

**coherent system**now known as

**Euclidean geometry**. For centuries scientists have read and reread the

*Elements*, which were fundamental to the progress of geometry and also influenced other scientific disciplines. Furthermore, from the interpretation of the

*Elements*in the contemporary age the

**non-Euclidean geometries**.

## Who was Euclid in brief?

Euclid lived in Alexandria, which in the age **Hellenistic**between the death of Alexander the Great (323 BC) and the Roman conquest of Egypt (30 BC), was the**undisputed cultural capital of the world**. The main cultural centers of the time were located in the city: the **museum**that is, “temple of the muses”, which was a sort of university (from which, moreover, our word “museum” derives, which however has a different meaning), and the **library**who owned almost all the books written up to that point.

Thanks to its cultural institutions, Alexandria was the main meeting point between the **Greek rational thought and oriental wisdom**from the combination of which scientific advances of fundamental importance for the development of civilization resulted.

Euclid was therefore located in the main cultural center of the world. However, we know very little about his life. His birthplace is not known but, according to ancient sources, he was **Plato’s youngest student**, the great Athenian philosopher. We also know that he lived in Alexandria approximately **between 320 and 270 BC. C.**and was one of the teachers at the museum.

Working in Alexandria, Euclid had the opportunity to regularly visit the library and have almost all the scientific knowledge of the time “at hand”.

## What is Euclid remembered for: the *Elements*

Euclid is best known as the author of the *Elements*a work that summarizes in the form of **definitions and propositions **(i.e. axioms that cannot be questioned) the mathematical knowledge existing at his time. Euclid, however, did not limit himself to summarizing what the other scholars had written: he corrected the wrong postulates, clarified the unclear ones better and added new ones. In doing so, he created a coherent geometric system, now known as **geometry ****Euclidean**.

The treatise of the Elements is divided **in thirteen books **(roughly equivalent to our chapters) and contains a total of 131 definitions and 465 propositions, all based on **five fundamental postulates**. The first four, easily understandable, are the following:

- One and only one straight line passes through two points.
- A straight line can be extended infinitely.
- Given a point and a length, a circle can be described.
- All right angles are congruent with each other (that is, they have the same shape and dimensions).

**The fifth postulate of ***Elements* and non-Euclidean geometries

*Elements*and non-Euclidean geometries

Euclid’s fifth postulate is the one he created **greater problems of interpretation**. In simple terms, it can be summarized with the following formula: “Given a line and a point external to it, there exists a unique parallel line passing through said point”.

For centuries scholars have tried to **connect the fifth postulate to the first four**but in the 19th century it was demonstrated that it is **completely independent**. From this discovery came the **non-Euclidean geometries**such as the spherical and hyperbolic ones, which do not accept one or more postulates.

In a nutshell, the main difference between Euclidean and non-Euclidean geometry lies in **concept of parallel line**: in Euclidean geometry there is always a straight line that passes through a point, parallel to another straight line present in the same plane; in non-Euclidean geometries, such a line could not exist.

**The fortune of ***Elements*

*Elements*

The Elements had an enormous diffusion and around the 1st century BC. C. established themselves as the main mathematics book of Greco-Roman civilization. In the following centuries they were translated into many languages, they became the** main “textbook” for the study of geometry** and they also influenced the other sciences.

The Elements had influence **even in politics**, because they presented a model of a system ordered and governed by immutable laws, which could also be applied to society. Among others, Abraham Lincoln, the American president of the Civil War years, was a passionate reader of Euclid’s work.

**What Euclid’s theorems state**

The current geometry books are named after Euclid **theorems**, derived from the eighth proposition of the VI book which, in the Italian translation of the mathematician Federico Enriques, is the following: “If in a right triangle the perpendicular is drawn from the right angle to the base, the triangles thus formed will be similar to the given one, and similar to each other.” The two theorems derived from the proposition define the relationships between the sides of a right-angled triangle.