(? ? -? , IIIe front S.J. - C.
Founder of the mathematical school
Greek mathematician. In Eukleidês Greek. One knows practically nothing of his person, if not that he lived a little before Archimedes, in the new city of Alexandria. Undoubtedly he was the founder of the mathematical school which illustrated this city. Perhaps its principal work, the Elements, results from a collaboration, but is certainly one of the books which particularly marked the history of the thought.
The Elements were known in Occident by Arab translations. They expose about all that was known in mathematics at the time of their drafting, with a systematic preoccupation with a rigor and an undeniable prevalence of the geometry on the arithmetic one. Their influence on mathematics was immense. It extended even beyond as testifies some by its structure and its style a work such as Principia of Isaac Newton.
Elements
The Elements are divided into thirteen books. The contents are presented in the form of a succession of proposals. They are sometimes theorems, stating truths and asking to be proven; sometimes of problems, calling a resolution. The statement itself is followed by the demonstration, or of the resolution according to the case. The concepts to which the proposals relate receive a definition. At the top of the first book the definitions of the simplest objects of the geometry appear: the point, the line, etc One also finds there the postulates and the axioms. The latter are “concepts common”, kinds of very general rules, applicable to the numbers as with the geometrical magnitudes. The postulates, five, are the statements of which it is asked to admit the validity so that the demonstrations have a starting point. The fifth postulate is equivalent saying that by a point it passes a parallel and only one on a given line. Its character of less obviousness induced research to try to show it starting from the four first. It is failure of these attempts which were born the not-Euclidean geometries.
Thirteen books
The first books of the Elements relate to the plane geometry. The book V exposes a theory of the proportions of a width and a rigor such as it did not have an equivalent before the XIX E century. It relates to the reports of the sizes. Those can be geometrical (lengths, surfaces, etc), as they can be arithmetic (numbers) or even still of different nature. Euclide differentiates the reports from sizes into commensurable (or rational) and incommensurable (or irrational).
Book VI describes the application of this theory to the similar figures. The three following relates to the numbers, tenth is devoted to the irrational sizes and the last to the solid geometry.
Book XIII, in particular, exposes the construction of the five regular solids (pyramid, octahedral, cubic, icosahedral and dodecahedron). Euclide wrote some other works, in geometry, optics and music.