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Heritage of Euklides

2005/12/01 Roa Zubia, Guillermo - Elhuyar Zientzia Iturria: Elhuyar aldizkaria

Most books written two thousand years ago have lost the present, but not all. The Greek mathematician Euklides wrote Elements, for example, no. The basis of the geometry taught today is contained in this ancient book. For this reason, for years it has been used as a textbook worldwide. It has been translated into many languages over the centuries and can now also be read in Basque. The Elhuyar Foundation has published the translation by the mathematician Patxi Angulo.
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Heritage of Euklides
01/12/2005 | Roa Zubia, Guillermo | Elhuyar Zientzia Komunikazioa
(Photo: I. Larrañaga)

The book Elements of Euklides has a noteworthy mark: it is the most translated and published scientific book. As he wrote about three hundred years before the birth of Christ, he is older than all the books that make up the New Covenant of the Bible.

At the time of its writing, Alexandria was a very new city, even from the point of view of traditions; Alexander the Great, founder, died a few years earlier and a very special atmosphere arose in a new growing city. Among other things, a tendency arose to gather the wisdom that existed until then, when the famous museum and library were launched. In that environment Euklides wrote the book Elements, probably with the intention of creating a collection of mathematical advances of the civilized world. It was not the only one, because there was a whole school of mathematicians in Alexandria (Apollonius's work is also remarkable), but the book Elements is the one that has most influenced the world of science.

"It can be said that the work has covered the person. The elements have long been known, but we know less about the person," says Patxi Angulo, a mathematician who has translated the book into Basque.

And it is true. On the one hand, because of the importance of the book and, on the other, because we know little about Euclid himself. He lived in Alexandria and worked in the museum, where he wrote all his works, we know nothing more. Some say that it did not exist and that Euklides was not a person, but a school, as with Pythagoras.

Collection of books

Page 1, 1768, of the first Portuguese version of Joameto Angelo Brunelli.
P. Angle

Being a man or a school, Euklides left a beautiful collection of mathematics, especially geometry. Elements is a collection of books. "The Greeks used to write the books like this," says Angulo. There are thirteen books that make up Elements, and each separately may not be enough to form a book. "In the Basque version we have completed some five hundred pages, but there are books of less than twenty pages."

In history, when the book has been translated or versions have been made, thirteen have not always been used. The first four, fifth and sixth, and the eleventh and twelfth are the most published. Surely they are the most practical and useful. But, according to Angulo, in addition to practicality, there may be other reasons for not using all chapters or books. "This also has to do with religion. For things to be well and well, there are parts that are not suitable from a religious point of view; they are too abstract and esoteric."

Perhaps, seen from current thinking, it is difficult to understand why, since the collection of books is full of very basic concepts of mathematics.

In the first four books and in the sixth the geometry of the plane is analyzed; in the fifth the proportions are analyzed; in books seven, eight and nine the theory of numbers is worked (the properties of numbers, for example); in the tenth the irrational numbers are analyzed; in the last three books, the geometry of space; in the eleventh and twelfth, the basic theorems are offered.

The thirteenth book is very special. Five regular polyhedra appear. But Plato took these polyhedra beyond current mathematical concepts, identifying regular polyhedra with elements of space: earth, sky, water, etc. Therefore, behind this idea there is a bit of mysticism, of perfection. For example, Plato said that the dodecahedron represents the whole universe, and that kind of thing. Perhaps, from the point of view of today's science, they are not very understandable things. But they are there.

Translation into Basque of the book Elements. It has been edited by the Elhuyar Foundation.

Euclidean life

However, most of the contents of the book Elements belong to the basis of geometry. And in our daily life we live in a euclidean space, in which we do any work at home, for example, all lines are perpendicular and parallel; we use triangles, circles and flat forms like these. All this is a euclidean geometry.

On the one hand, euclidean geometry studies plane geometry: triangles, squares, circles, Pythagorean theorem, direct parallel theorems of Thales, etc. On the other hand, it is included in the geometry of space, as cones, cylinders, spheres and relations appear between them.

"Another thing is how it is taught in school," says Angulo. "Some theorems or properties are taught, others pass or at least are not taught this way, or are not very important today. But basic mathematics is there, we learn in school."

Description

The book Elements is a kind of traces of ancient Greeks: it is a vestige of a very rich activity.
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We are not the only ones who have learned the geometry of Euclid in school; in educational texts it is a classic that has spread to many places and times through the book Elements. Sometimes the full text has been used, but in most cases content has been deleted or added. Or rather, removing and adding content, both together.

No wonder in a book written 2,300 years ago. In fact, it seems impossible for an unchanged version to remain so long. The original book was lost, but throughout history some people used the book, or parts of it. Although it was not the whole book, they transmitted their parts. Proof of this are Proklo and Teon Izmirn. In addition, the book was used by numerous Arab writers. The Arabs recovered the original texts, not the true papyri, but collected many manuscripts that, at least, were not lost.

The book was used in schools during the Middle Ages and in the XIX. Until the twentieth century it has also been a textbook, but moved to every age and type of society, adapted. XIX. However, an exhaustive study was conducted in the 20th century to identify the original text. In fact, the Danish Heiberg distinguished between what he himself had left and subsequent contributions. And he translated the material from Euklides into modern Greek. This work is based on what is left of the original text of Euklides. He distributed what was written by Euklides and later added it. There are always doubts; it seems that notes were added, even theorems that the original did not have, and that sometimes the explanations were extended or added. It is difficult to discern all of this, but today Heiberg's work is based on it.

Now, taking as reference the work of Heiberg, the mathematician Patxi Angulo has translated into Basque Elements. Therefore, the old book continues to advance through more translations and publications. It is possible that the book has a validity of 2,300 years more, what we know is that it remains a reference of the moment.

For example, the ecilateral triangle
Euclid uses elementary geometry postulates to make propositions. For example, in this proposition explains how to form a rotating triangle from a line:
(Photo: G. roa)
"Let AB be the finite line given. A rotating triangle must be built on the AB line. Circular BCD at distance A and AB. Repeat the ACE circle taking distance B and BA. And the straight CA, CB from point C to points A, B that are cut together."
Translating elements into Basque, a great job
As for the book Elements, it is currently based on the work done by the Danish Heiberg. He separated those written by Euclid himself from the parts that have subsequently been added or removed. Based on this work, UPV-EHU mathematics professor Patxi Angulo has translated the book from the two translations of Heiberg's version, and has used four other versions that helped him resolve his doubts.
"I haven't relied on that job because I don't know the Greek," says Angulo. "Heiberg wrote in Greek, but then many other translations have been based on that work, and precisely the six translations I have used are based on Heiberg." Two are in Spanish, two in English and two in French.
A Spanish version translated by María Luisa Puertas was published in 1996. One English is from 1908, Thomas L. English heath. A Frenchman was published by Bernard Vitrac in 2001. "I used this third party especially to make some corrections and resolve some doubts." He also used three others.
Patxi Angulo.
(Photo: n. Hardware)
"I contacted Greek teachers (Cristina Lasa and Javier Alonso). They have helped me understand the Greek words. In addition, I have had the collaboration of two friends, one as an advisor, Xabier Artola, to help them decide before the doubts that arise at each moment and a corrector of the whole text, José Ramón Etxebarria".
The book has been translated into several languages. Available in Italian, German, French, Dutch, English, Spanish, Russian, Swedish, Danish, Modern Greek, Catalan, Japanese and Portuguese. He appeared in Catalan two years ago, in 2003. Now, of course, we also have it in Basque. Also, of course, there were ancient versions such as Arabic and old Greek.
The Basque language has played Angulo a lot. He has worked for two years. "It has been a beautiful job and in both ways: robust and beautiful," he says happy, with the book in his hands.
Triangles away from the hand of Euklides
(Photo: G. roa)
In the flat world of Euclid, the sum of the angles of a triangle gives rise to 180 degrees. But it is not so in all geometries. Observe, for example, this triangle: it has a vertex in the North Pole and the other two in the equator, one in the Greenwich meridian and another in the west longitude of 90 degrees. In this triangle each angle has 90 degrees and the sum of three is 270 degrees. It shows that spherical geometry is not euclidean (not flat). And it's not just spherical. Einstein's Theory of Relativity is exposed in a hyperbolic geometry. This geometry is also non-euclidean.
Roa Bridge, Guillermo
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2005
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Mathematics; History