Euclid's Elements

The Elements of Euclid (Greek: Στοιχεῖα, / stoicheia/) and known as Euclidean geometry; in Greek: Ευκλειδης Γεωμετρια) is a mathematical and geometric treatise that consists of thirteen books, written by the Greek mathematician and geometer Euclid around 177 BC. C. in Alexandria. Through these books the author offers a definitive treatment of the geometry of two dimensions (the plane) and three dimensions (space).

History

Although the work was known in Byzantium, it was unknown in Western Europe until around 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation. In 1482, Erhard Ratdolt made the first Latin printing of the work in Venice.

The Elements is considered one of the most popular textbooks in history and the second in number of editions published after the Bible (more than 1000). For several centuries, the quadrivium was included in the syllabus of university students, and knowledge of this text was required. It is still used today by some educators as a basic introduction to geometry.

In these thirteen volumes Euclid compiles a large part of the mathematical knowledge of his time, represented in the axiomatic system known as Euclid's Postulates, which in a simple and logical way give rise to Euclidean Geometry.

Fundamental principles

In the first book, Euclid develops 48 propositions from 23 definitions (such as point, line, and surface), 5 postulates, and 5 common notions (axioms). Among these propositions is the first known general proof of the Pythagorean theorem.

The common notions of The Elements are:

1. Things equal to the same thing are equal to each other.
2. If they add equal to equals, all are equal.
3. If they are equal to equal, the remains are equal.
4. The things that match one another are the same with each other.
5. Everything is greater than the part

The postulates of The Elements are:

1. A straight line can be drawn by joining any two points.
2. A straight line segment can be extended indefinitely in a straight line.
3. Given a straight line segment, a circle can be drawn with any center and distance.
4. All straight angles are the same.
5. Postulate of the parallels. If a straight line cuts to another two, so that the sum of the two inner angles on the same side is less than two straight, the other two straights are cut, by prolonging them, on the side on which are the minor angles than two straight.

This last postulate has an equivalent, which is the most used in geometry books:

• From an outer point to a straight line, a single parallel can be traced

Note that this is the postulate that makes geometry Euclidean. Negating it, non-Euclidean geometries are obtained.

These basic principles reflect Euclid's interest in constructive geometry, as do contemporary Greek and Hellenistic mathematicians.

Content

A fragment of the Elements of Euclides found in Oxirrinco, dated to the year 100 a. C.
The diagram accompanies Proposition 5 of Book II.

Despite being a work on geometry, the book includes results that today can be classified within number theory. Euclid decides to describe the results in number theory within geometry because he was unable to develop a constructive approach to arithmetic.

The contents of the books are as follows:

• Books 1 to 4 deal with flat geometry.
• Books 5 to 10 deal with reasons and proportions.
• Books 11 to 13 deal with solid body geometry.

Euclid's Elements / Definitions (First Book)

Cover of the work Geometric Elements of Euclides published by Jacobo Kresa in 1689

1. A point is what has no parts.
2. A line is a length without width.
3. The ends of a line are points.
4. A straight line is the one that lies equally with the points that are in it.
5. A surface is what only has length and width.
6. The ends of a surface are lines.
7. A flat surface is the one that lies equally with the lines that are in it.
8. A flat angle is the mutual inclination of two lines that meet one another in a plane and are not in a straight line.
9. When the lines that understand the angle are straight, the angle is called rectilinear.
10. When a straight line raised on another straight form equal adjacent angles, each of the equal angles is straight and the raised is called perpendicular to the one on which it is.
11. Obtuse angle is the greater than a rectum.
12. Acute angle is the lower than a rectum.
13. A limit is what is extreme of something.
14. One figure is the contents by one or more limits.
15. A circle is a flat figure covered by a single line so that all the drawn lines falling on it from a point of those within the figure are equal to each other.
16. The point is called the "center" of the circle.
17. A diameter of the circle is any straight drawn through the center and limited in both senses by the circumference of the circle, which also divides the circle into two equal parts.
18. A semi-circle is the figure between the diameter and the circumference by it cut. And the center of the semi-circle is the same as that of the circle.
19. Rectilinear figures are those covered by straight lines, trilaterals that are covered by 3, quadrilaterals that are covered by 4, multilaters that are covered by more than 4 straights.
20. Among the trilateral figures, the equilateral triangle is the one that has the three equal sides, isosceles triangle which has two equal sides, and the escalen triangle which has the three unequal sides.
21. Among the trilateral figures, triangle rectangle is the one that has a straight angle, obtussangle the one that has an obtuso angle, acungulo which has the three sharp angles.
22. Of the quadrilateral figures, square is the one that is equilateral and rectangular, rectangle the one that is rectangular but not equilateral, rombo the one that is equilateral but not rectangular, romboide the one that has the opposite angles and sides equals between itself, but not equilateral or rectangular; and trapecios the other quadrilateral figures.
23. They are parallel straights that being on the same plane and being prolonged indefinitely in both senses, are not found one to another in any of them.

Euclid's Elements / Postulates (First Book)

1. Postulate a straight line from any point to any point.
2. And continually prolonging a fine line in a straight line.
3. And describing a circle with any center and distance.
4. And being all straight angles equal with each other.
5. And if a straight inciding on 2 straights makes the inner angles on the same side less than two straight, the two prolonged straights indefinitely will be on the side on which the minors are the two straight.

Basic Notions (Book One)

1. If two elements are equal to another element, they are equal to each other.
2. If two equal amounts are added to other equals, they will be equal.
3. If two equal amounts are subtracted to other equals, they will be equal.
4. If two elements match one with another, they're the same.
5. Everything is greater than a part.

Transmission and translations into Spanish

Euclid's text was transmitted in two ways. Copies were made in Christian countries, especially during the Byzantine Empire, where knowledge of Greek was more widespread, but the study of geometry was not widely disseminated in this area. On the other hand, in the Arab world the Elements were translated and commented on and a more lively tradition was maintained, although less faithful to Euclid. The main translators into Arabic were Al-Hajjàj (fl. 786-833) and Ishàq-ben-Hunayn (9th century). The Hispanic Muslim Avicenna wrote a commentary on the Elementos. The best Latin translation of the Middle Ages was that of Gerardo de Cremona (1114-1187), Italian by origin but settled in Toledo, of whose Cathedral he became a canon, from the Arabic version of Thábit ibn Qurra. But the most popular Latin version during the Late Middle Ages was that of Campano de Novara (13th century); in fact, it was this version that was first printed (Venice, 1482).

The first version in Spanish was translated by Rodrigo Zamorano (Seville, 1576). They were followed by that of Luis Carduchi (1637), that of Andrés Puig (1672), that of José Zaragoza (1678), that of Sebastián Fernández de Medrano (1688), the commented on by the Czech Jesuit mathematician Jacobo Kresa (Elementos geométricos de Euclid, the first six books of plans, the eleventh and dozenth of solids: with some select Theorems of Archimedes, Brussels, 1689), that of Francisco Larrando de Mauleón (1698), that of Pedro de Ulloa (1706), that of the novator Tomás Vicente Tosca (1707), that of Antonio José Deu y Abella (1723), those of Gaspar Álvarez (1739) and Blas Martínez de Velasco (1747), the translation of Robert Simson (1774), that of Juan Justo García (1782), that of Pedro Giannini (1788) and the already very numerous ones from the 19th and 20th centuries. Of the old ones, only the first edition, prepared by Zamorano, and one of the last ones, the Spanish translation of Robert Simson's version, try to be a faithful transcription of the Greek text. Most are educational adaptations, more or less free, of the work of Euclid. According to Juan Navarro Loidi,

The Renaissance or Illustration versions have almost all the same books. They usually include books from I to VI that deal with flat geometry, although book V has often been simplified and converted into a manual on numerical fractions applied to geometry. From the edition of Zaragoza (1678) they also contain the first two books of space geometry. But, in Book XII, simplifications are very common to avoid long demonstrations by the method of “exhaustion”. They are usually inspired by the idea of limit of figures introduced by the Belgian Tacquet, although Zaragoza is inclined by the infinite sums, perhaps influenced by Cavalieri.

Modern editions are those of Juan David García Bacca, Euclid. Complete works. Geometry Elements (Mexico, 1954-1956), which actually contains only books I to V of the text. Francisco Vera, in volume I of his Greek Scientists (Madrid, 1970) published the work of Euclid with all his books from I to XIII for the first time in Spanish; but in that edition, especially in book X, the demonstrations are greatly abbreviated or replaced by more modern ones, so it is not really a translation of the complete text of Euclid. The first complete Spanish translation of the Elements is that of María Luisa Puertas Castaños Elementos (Madrid 1991, 1994 and 1996) in three volumes.

Bases on previous works

An enlightenment of a manuscript based on the translation of Elements of Adelardo de Bath, c. 1309-1316; that of Adelard is the oldest translation that is preserved from the Elements to Latin, made in the work of the 12th century and translated from Arabic.