# Geometry

**Geometry** is a branch of mathematics that studies shapes, both in the plane and in space, and how they relate to each other.

The word "geometry" originates from the Latin "geometrĭa" and the ancient Greek "γεωμετρία," which literally means "measurement of the earth."

**Note**. It combines "Geo" (γῆ), meaning "earth," and "metria," a Greek word meaning "measurement." Thus, geometry is essentially the study of the measurement and relationships of shapes on the earth and in space.

## The History of Geometry

The earliest studies date back to ancient civilizations, who used it to solve practical problems. For instance, the ancient Egyptians used geometry to redraw the boundaries of fields after the Nile's floods.

**Note**. The annual floods of the Nile altered the extent and boundaries of agricultural fields. Therefore, it was necessary to periodically map the boundaries using precise measurement techniques, also for taxation purposes. Hence, the term "geometry," which literally means "measurement of the earth."

The Babylonians mainly used geometry for star observation, while the Phoenicians used it for navigation.

In the 6th century BC, the Greeks began to approach geometry more theoretically and independently of practical applications, viewing geometric figures as **abstract models** of the real world to be analyzed logically.

This approach marks the beginning of **rational geometry**.

The Greeks sought to organize and rationalize geometric knowledge through logical reasoning, deductions, and proofs.

Instruments like the straightedge and compass became widely used, leading to new demonstrative techniques. Greek geometry became fundamental to the development of various sciences such as geography, astronomy, optics, and mechanics, as well as many practical techniques like navigation.

Many Greek philosophers and mathematicians, such as Thales, Pythagoras, Plato, Aristotle, and Euclid, significantly contributed to this field.

In particular, Greek geometry is closely associated with Euclid, a Greek mathematician of the 3rd century BC, because he compiled all the geometric knowledge of his time into a work called "**Euclid's Elements**."

This book, very detailed and precise, was for centuries the standard text for the study of geometry and mathematics. Even today, it is one of the first topics covered in school geometry books.

**Note**. Euclid's work is also one of the earliest examples of an axiomatic system in geometry because his method was based on principles accepted as true without proof, called axioms, from which he derived all other geometric theorems.

## Rational or Euclidean Geometry

Rational geometry, also known as **Euclidean geometry**, is a type of geometry that follows well-defined rules.

These rules, which include definitions, primitive entities, postulates, and theorems, were first systematized by the Greek mathematician Euclid in the 3rd century BC.

For this reason, we often refer to this discipline as "Euclidean geometry."

**Note**. Essentially, rational geometry is a structured study that helps us understand the nature of shapes and dimensions in our world, based on the fundamental rules established by Euclid.

In his first book "Elements," Euclid lays the foundation of his work with 23 definitions explaining what a point, a line, and a surface are.

Additionally, he presents 5 postulates and 5 "common notions," which are basic rules.

The "**common notions**" are as follows:

- If two things are equal to the same thing, then they are equal to each other.
- If you add the same amount to two equal things, the sums will be equal.
- If you subtract the same amount from two equal things, the remainders will be equal.
- If two things coincide with the same thing, then they are equal to that thing.
- A whole is always greater than its part.

The **five postulates of Euclid** are as follows:

- You can draw a straight line between any two points.
- You can extend a straight line segment infinitely.
- You can draw a circle with any straight line segment as the radius and one endpoint of the segment as the center.
- All right angles are equal to each other.
- If you draw two lines that intersect a third line so that the sum of the internal angles on one side is less than two right angles, then the two lines will intersect on the same side if extended far enough.

## Analytic Geometry

**Analytic geometry** links geometry and algebra through the use of coordinate systems to represent points and shapes in space.

It allows the description of geometric figures using equations and facilitates the calculation of distances, angles, and intersections.

This approach transforms complex geometric problems into solvable algebraic problems, innovatively merging two fundamental branches of mathematics.