EUCLIDEAN GEOMETRY: HISTORY, BASIC CONCEPTS, AND EXAMPLES - MATHS - 2025

The Euclidean geometry corresponds to the study of the properties of geometric spaces where Euclid's axioms are satisfied. Although this term is sometimes used to cover geometries that have higher dimensions with similar properties, it is generally synonymous with classical geometry or plane geometry.

In the III century a. C. Euclides and his disciples wrote the Elements, a work that encompassed the mathematical knowledge of the time endowed with a logical-deductive structure. Since then, geometry became a science, initially to solve classical problems and evolved to be a formative science that helps reason.

History

To talk about the history of Euclidean geometry, it is essential to start with Euclid of Alexandria and the Elements.

When Egypt was left in the hands of Ptolemy I, after the death of Alexander the Great, he began his project in a school in Alexandria.

Among the sages who taught at the school was Euclid. It is speculated that his birth dates from approximately 325 BC. C. and his death of 265 a. C. We can know with certainty that he went to Plato's school.

For more than thirty years Euclid taught in Alexandria, building its famous elements: he began to write an exhaustive description of the mathematics of his time. Euclid's teachings produced excellent disciples, such as Archimedes and Apollonius of Perga.

Euclid was in charge of structuring the disparate discoveries of the ancient Greeks in the Elements, but unlike his predecessors he does not limit himself to affirming that a theorem is true; Euclid offers a demonstration.

The Elements is a compendium of thirteen books. After the Bible, it is the most published book, with more than a thousand editions.

Euclid's Elements

The Elements is Euclid's masterpiece in the field of geometry, and offers a definitive treatment of the geometry of two dimensions (the plane) and three dimensions (space), this being the origin of what we now know as Euclidean geometry.

Basic concepts

The elements are made up of definitions, common notions and postulates (or axioms) followed by theorems, constructions and proofs.

- A point is that which has no parts.

- A line is a length that has no width.

- A straight line is one that lies equally in relation to the points that are in it.

- If two lines are cut so that the adjacent angles are equal, the angles are called straight lines and the lines are called perpendicular.

- Parallel lines are those that, being in the same plane, never intersect.

After these and other definitions, Euclid presents us with a list of five postulates and five notions.

Common notions

- Two things that are equal to a third are equal to each other.

- If the same things are added to the same things, the results are the same.

- If equal things are subtracted equal things, the results are equal.

- Things that match each other are equal to each other.

- The total is greater than a part.

Postulates or axioms

- One and only one line passes through two different points.

- Straight lines can be extended indefinitely.

- You can draw a circle with any center and any radius.

- All right angles are equal.

- If a straight line crosses two straight lines so that the interior angles of the same side add up to less than two right angles, then the two lines will cross on that side.

This last postulate is known as the parallel postulate and was reformulated in the following way: "For a point outside a line, a single parallel to the given line can be drawn."

Examples

Next, some theorems of the Elements will serve to show properties of geometric spaces where the five postulates of Euclid are fulfilled; In addition, they will illustrate the logical-deductive reasoning used by this mathematician.

First example

Proposition 1.4. (LAL)

If two triangles have two sides and the angle between them is equal, then the other sides and the other angles are equal.

Demonstration

Let ABC and A'B'C 'be two triangles with AB = A'B', AC = A'C 'and the angles BAC and B'A'C' equal. Let's move triangle A'B'C 'so that A'B' coincides with AB and that angle B'A'C 'coincides with angle BAC.

So line A'C 'coincides with line AC, so that C' coincides with C. Then, by postulate 1, line BC must coincide with line B'C '. Therefore the two triangles coincide and, consequently, their angles and their sides are equal.

Second example

Proposition 1.5. (

Suppose triangle ABC has equal sides AB and AC.

So, the triangles ABD and ACD have two equal sides and the angles between them are equal. Thus, by Proposition 1.4, the angles ABD and ACD are equal.

Third example

Proposition 1.31

You can construct a line parallel to a line given by a given point.

Building

Given a line L and a point P, a line M is drawn through P and intersects L. Then a line N is drawn through P that intersects L. Now, a line N is drawn through P, which intersects M, forming an angle equal to the one that L forms with M.

Affirmation

N is parallel to L.

Demonstration

Suppose that L and N are not parallel and intersect at a point A. Let B be a point in L beyond A. Consider the line O that passes through B and P. Then, O intersects M at angles that add up to less than two straight.

Then by 1.5 line O must intersect line L on the other side of M, so L and O intersect at two points, which contradicts Postulate 1. Therefore, L and N must be parallel.

References

1. Euclid. Elements of Geometry. National Autonomous University of Mexico
2. Euclid. The first six books and the eleventh and twelfth of Euclid's elements
3. Eugenio Filloy Yague. Didactics and history of Euclidean geometry, Grupo Editorial Iberoamericano
4. K. Ribnikov. History of Mathematics. Mir Editorial
5. Viloria, N., & Leal, J. (2005) Plane Analytical Geometry. Editorial Venezolana CA