EUCLIDES: BIOGRAPHY, CONTRIBUTIONS AND WORK - SCIENCE - 2024

Euclid of Alexandria was a Greek mathematician who laid important foundations for mathematics and geometry. Euclid's contributions to these sciences are of such importance that they are still valid today, after more than 2000 years of being formulated.

This is why it is common to find disciplines that contain the adjective "Euclidean" in their names, since they base part of their studies on the geometry described by Euclid.

Euclid, 300 BC

Biography

The exact date on which Euclid was born is not known. Historical records have allowed his birth to be located sometime near 325 BC.

Regarding his education, it is estimated that it took place in Athens, due to the fact that Euclid's work showed that he knew in a deep way the geometry that was generated from the Platonic school, developed in that Greek city.

This argument holds until it follows that Euclid did not seem to know the work of the Athenian philosopher Aristotle; For this reason, it cannot be conclusively stated that Euclid's formation was in Athens.

Teaching work

In any case, it is known that Euclid taught in the city of Alexandria when King Ptolemy I Soter, who founded the Ptolemaic dynasty, was in command. It is believed that Euclides resided in Alexandria around 300 BC, and that he created a school there dedicated to the teaching of mathematics.

During this period, Euclides gained considerable fame and recognition, as a result of his skill and gifts as a teacher.

An anecdote related to King Ptolemy I is the following: some records indicate that this king asked Euclid to teach him a quick and summary way of understanding mathematics so that he could apprehend and apply it.

Given this, Euclides indicated that there are no real ways to obtain this knowledge. Euclid's intention with this double meaning was also to indicate to the king that because he was powerful and privileged, he could not understand mathematics and geometry.

Personal characteristics

In general, Euclid has been portrayed in history as a calm person, very kind and modest. It is also said that Euclid fully understood the enormous value of mathematics, and that he was convinced that knowledge in itself is invaluable.

In fact, there is another anecdote about it that transcended our time thanks to the doxographer Juan de Estobeo.

Apparently, during a Euclid class in which the subject of geometry was discussed, a student asked him what was the benefit that he would find obtaining that knowledge. Euclides answered him firmly, explaining that knowledge by itself is the most invaluable element that exists.

As the student apparently did not understand or endorse his master's words, Euclides directed his slave to give him some gold coins, emphasizing that the benefit of geometry was much more transcendent and profound than a cash reward.

In addition, the mathematician indicated that it was not necessary to make a profit from each knowledge that was acquired in life; the fact of acquiring the knowledge is, in itself, the greatest gain. This was Euclid's view in relation to mathematics and, specifically, geometry.

Death

According to historical records, Euclid died in 265 BC in Alexandria, the city in which he lived much of his life.

Plays

The elements

Euclides' most emblematic work is The Elements, made up of 13 volumes in which he speaks on topics as varied as geometry of space, incommensurable magnitudes, proportions in the general sphere, plane geometry and numerical properties.

It is a comprehensive mathematical treatise that had great significance in the history of mathematics. Even Euclid's thought was taught until the 18th century, long after his time, a period in which the so-called non-Euclidean geometries arose, those that contradicted Euclid's postulates.

The first six volumes of The Elements deal with the so-called elementary geometry, there topics related to proportions and the techniques of geometry used to solve quadratic and linear equations are developed.

Books 7, 8, 9, and 10 are devoted exclusively to solving number problems, and the last three volumes focus on the geometry of solid elements. In the end, the structuring of five polyhedra in a regular way, as well as their delimited spheres, is conceived as a result.

The work itself is a great compilation of concepts from previous scientists, organized, structured and systematized in such a way that it allowed the creation of a new and transcendent knowledge.

Postulates

In The Elements Euclid proposes 5 postulates, which are the following:

1- The existence of two points can give rise to a line that unites them.

2- It is possible that any segment is continuously lengthened in a straight line without limits directed in the same direction.

3- It is possible to draw a center circle at any point and at any radius.

4- All the right angles are equal.

5- If a line that intersects two other lines generates angles smaller than the straight lines on the same side, these lines extended indefinitely are cut in the area where these smaller angles are.

The fifth postulate was made in a different way later: since there is a point outside a line, only a single parallel can be traced through it.

Reasons for significance

This work by Euclid had great significance for various reasons. In the first place, the quality of the knowledge reflected there caused that the text was used to teach mathematics and geometry at the basic education levels.

As mentioned above, this book continued to be used in academia until the 18th century; that is to say, it had a validity of approximately 2000 years.

The work The elements was the first text through which it was possible to enter the field of geometry; Through this text, deep reasoning based on methods and theorems could be carried out for the first time.

Second, the way in which Euclides organized the information in his work was also very valuable and transcendent. The structure consisted of a statement that was reached as a consequence of the existence of several principles, previously accepted. This model was also adopted in the fields of ethics and medicine.

Editions

As for the printed editions of The Elements, the first was produced in the year 1482, in Venice, Italy. The work was a translation into Latin from the original Arabic.

After this issue, more than 1000 editions of this work have been published. This is why Los elements has come to be considered one of the most widely read books in all of history, along with Don Quijote de la Mancha, by Miguel de Cervantes Saavedra; or even on a par with the Bible itself.

Main contributions

Elements

The most recognized contribution of Euclides has been his work entitled The elements. In this work, Euclides collected an important part of the mathematical and geometric developments that had taken place in his time.

Euclid's theorem

Euclid's theorem demonstrates the properties of a right triangle by drawing a line that divides it into two new right triangles that are similar to each other and, in turn, are similar to the original triangle; then, there is a relationship of proportionality.

Euclidean Geometry

Euclid's contributions were mainly in the field of geometry. The concepts developed by him dominated the study of geometry for almost two millennia.

It is difficult to give an exact definition of what is Euclidean geometry. In general, this refers to the geometry that encompasses all the concepts of classical geometry, not just Euclid's developments, although he collected and developed several of these concepts.

Some authors assure that the aspect in which Euclides contributed more to geometry was his ideal of founding it on an incontestable logic.

For the rest, given the limitations of the knowledge of his time, his geometric approaches had several shortcomings that later other mathematicians reinforced.

Demonstration and math

Euclides, along with Archimedes and Apolinio, are considered the perfectors of the proof as a chained argument in which a conclusion is reached while justifying each link.

The proof is fundamental in mathematics. Euclid is considered to have developed the processes of mathematical proof in a way that endures to this day and is essential in modern mathematics.

Axiomatic methods

In Euclid's presentation of geometry in The Elements, Euclid is considered to have formulated the first "axiomatization" in a very intuitive and informal way.

Axioms are basic definitions and propositions that do not require proof. The way in which Euclid presented the axioms in his work later evolved into an axiomatic method.

In the axiomatic method, definitions and propositions are posed so that each new term can be eliminated by previously entered terms, including axioms, to avoid infinite regression.

Euclides indirectly raised the need for a global axiomatic perspective, which led to the development of this fundamental part of modern mathematics.

References

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4. Florian C. Euclid of Alexandria and the Bust of Euclid of Megara. Science, New Series. 1921; 53 (1374): 414–415.
5. Hernández J. More than twenty centuries of geometry. Book Magazine. 1997; 10 (10): 28–29.
6. Meder AE What is Wrong with Euclid? The Mathematics Teacher. 1958; 24 (1): 77–83.
7. Theisen BY Euclid, Relativity, and sailing. Mathematical history. 1984; 11: 81–85.
8. Vallee B. The complete analysis of the binary Euclidean algorithm. International Algorithmic Number Theory Symposium. 1998; 77-99.