Elements of Euclide

See also: Element, Elements for European civilization

The Elements (in Greek old Στοιχεία / Stoikheía ) is a mathematical treaty and Géométrique, made up of 13 pounds organized thématiquement, probably written by the Greek mathematician Euclide towards 300 av. J. - C. It includes/understands a collection of definitions, Axiome S, Théorème S and them Démonstration on the subjects of the Euclidean Géométrie and the primitive Théorie of the numbers.

The Éléments are the oldest known example of a treatment axiomatic and systematic of the geometry and its influence on the development of logic and Western science is fundamental. It is probably about the collection which met the most success during the History: the Éléments were one of the first printed books (Venice, 1482) and is preceded only by the Bible for the number of editions published (largely more than 1.000). During centuries, it belonged to the standard university course.

Principles

The method of Euclide consisted in basing its work on ordinary definitions, postulates and “concepts” (these two terms would nowadays be called Axiome S). For example, the delivers I contains 23 definitions (not, Ligne, Surface, etc), five postulates and five concepts ordinary.

Postulates of book I:

a segment of right-hand side can be traced by uniting two unspecified points.
a segment of right-hand side can be prolonged indefinitely in a straight line.
being given an unspecified segment of right-hand side, a Cercle can be traced by taking this segment like ray and one of its ends like center.
All the right angles are congruent S.
If two lines are secant with a third in such way that the sum of the interior angles of an east coast lower than two right angles, then these two lines are inevitably secant on this side.

Ordinary notions of book I:

Of the things which are equal to the same thing is equal between them.
Si of the equal things is added to other equal things, theirs sums are equal.
Si of the equal things is withdrawn from other equal things, the remainders are equal.
Of the things which coincide with another is equal between them.
the whole is larger than the part.

Posterity

The success of the Éléments is due mainly to the logical presentation of the near total of the mathematical knowledge Euclide had. The systematic and effective use of the development of the demonstrations starting from a reduced type font of axioms encouraged to use them as delivers reference during centuries.

Throughout the History, some controversies surrounded the axioms and the demonstrations of Euclide. Nevertheless, the Éléments remain a fundamental work in the history of sciences and were of a considerable influence. The European scientists Nicolas Copernic, Johannes Kepler, Galileo Galilei and particularly Isaac Newton all were influenced by the Éléments and applied their knowledge of the book to their clean work. Certain mathematicians (Bertrand Russell, Alfred North Whitehead) and philosophers (Baruch Spinoza) also tried to write their clean Éléments , of the axiomatic deductive structures applied to their respective disciplines.

In five postulates stated in the I, the last delivers, which one deduces the postulate from the parallels: “in a point external on a line, only one single line passes which is parallel for him”, always seemed less obvious than the others. Several mathematicians suspected that it could be shown starting from the other postulates, but all the attempts with this intention failed. About the middle of, it was shown that such a demonstration does not exist, that the fifth postulate is independent of the four others and that it is possible to build not-Euclidean geometries coherent by taking its negation.

History

The first hard copies of the concepts lengths and of orthogonality are Babylonian and go back to one period located between 1900 and 1600 av. J. - C.. One finds there the knowledge of the Théorème of Pythagore at least in the case of a Triangle whose with dimensions ones are respective lengths three, four and five.

The first formalization is gathered in a book called the the Elements. It contains all the mathematical knowledge of the time. Although the majority of the theorems are former for them, the Éléments were sufficiently complete and rigorous to eclipse geometrical works which preceded them and little of things are known on the pre-Euclidean geometry.

Its author Euclide of Alexandria (325 - 265 av. J. - C.) is a Greek mathematician who was probably a disciple of Plato (348 - 265 av. J. - C.) . Its history as that of this book are badly known. Three assumptions are advanced about it. Euclide is:

* either a historical character principal author of the Elements ,

* or the head of a mathematical school

* or a name of author whom a group of mathematicians used to write a compilation, this name would be then a reference to the Greek philosopher Euclide de Mégare (450 - 380 av. J. - C.) .

If the first assumption were allowed without the shade of a doubt during more than 2000 years, it remains still most probable. On the other hand, it is practically established that Euclide was with the head of a vigorous mathematical school and its disciples contributed certainly to the drafting of the Eléments . Hippocrates de Chios (470 - 410 av. J. - C.) is the author of the contents of books I and II of the elements, if one believes the Byzantine philosopher of it Proclos (411 - 487) . He writes of him “He was the first to be written for now known compilation under the name of the Eléments ”.

The work was translated into Arab after being given to the Arabs by the Byzantine Empire, then translated into Latin according to the Arab texts (Adelard de Bath at the 12th century, taken again by Campanus de Novare). The first printed edition goes back to 1482 and it book since was translated in a multitude of languages and was published in more than 1.000 different editions. Copies of the Greek text always exist, for example in the Bibliothèque of the Vatican or with the Bodleian Library in Oxford, but these manuscripts are of variable quality and always incomplete. By analysis of the translations and originals, it was possible to put forth assumptions on the original contents, of which there remains no integral copy.

Later axiomatization

The mathematicians of discovered that the demonstrations of Euclide require additional assumptions, not specified in the original text. David Hilbert modified the list to provide a complete set of it in 1899 in an article entitled the bases of the geometry . The list of the Axiomes of Hilbert contains 20 of them.

Books

The Éléments are organized as follows:

books I to IV treat plane geometry:

the delivers I states the basic properties of the geometry: Theorem of Pythagore, equalities angular and of surface S and parallelism, nap of the angles of the Triangle, three Congruence of triangles.
the delivers II is usually named delivers geometrical Algèbre, because it is a book of geometry easy to interpret like algebra, which it is not exactly but it was included/understood and used in Arab mathematics for the algebra.
the delivers III milked Cercle and of its properties: Inscribed angle, power of a point, tangent.
the delivers IV deals of the inscription and the district of Triangle S or regular polygons in the Cercle.

the books V with X utilize the proportions:

the delivers V is the treaty of the proportions of sizes.
the delivers VI is that of the application of the proportions to the geometry: Theorem of Thalès, figures similar.
the delivers VII is devoted to the Arithmétique: divisibility, prime numbers, PGCD, PC.
the delivers VIII milked Arithmétique of the proportions and geometrical continuations.
the delivers IX applies the precedents: infinity of the prime numbers, nap of a geometrical series, perfect numbers.
the delivers X is an attempt at classification of the irrational sizes introducing the method by exhaustion, which precedes the integration, irrationality of $\ sqrt {2}$ .

books XI to XII treat solid geometry:

the book XI generalizes in space books I to VI: perpendicularity, parallelism, Volume S of Parallélépipède S.
the delivers XII calculates surfaces and volumes by using the method of exhaustion: disc, cone S, Pyramid S, Cylinder S and Sphere.
the delivers XIII is the generalization of book IV in space: gilded Section, five polyhedral regular registered in a sphere.

There exist two books apocryphal books, present in appendix in the translation of Heath.

See too

External bonds

Euclide. Fifteen books of the geometric standards of Euclide: more the book of give mesme Euclide also traduict as a François by the aforementioned Henrion, and printed of alive sound, translation of 1632, site Gallica
'' Euclid' S Elements '' adapted for Internet by D.E. Joyce
'' Oliver Byrne' S edition off EUCLID '' (version colors)

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