Euclides: Libros XI-XIII-XIII

Abstract

En este trabajo de Fin de Grado estudiamos los tres ´ultimos libros de los Elementos de Euclides, que est´an dedicados a la geometr´ıa del espacio. El Libro XI comienza con un conjunto de 28 definiciones que valen para los tres libros: s´olidos, rectas y planos ortogonales, planos paralelos, ´angulos s´olidos, s´olidos semejantes, pir´amide, prisma, esfera, cono, cilindro y los cinco poliedros regulares. La mayor parte de este libro, que cuenta con 39 proposiciones, se centra en el an´alisis pormenorizado de los s´olidos paralelep´ıpedos. En el Libro XII, a trav´es de 18 proposiciones, relaciona los vol´umenes de prismas y pir´amides, de cilindros y conos, y de la esfera con el cubo de su di´ametro, mediante una magistral utilizaci´on de la teor´ıa de la proporcionalidad y del m´etodo de exhauci´on. Finalmente, el principal objetivo del Libro XIII y un digno colof´on a esta magnum opus de Euclides, es la construcci´on de los poliedros regulares o s´olidos plat´onicos: tetraedro, octaedro, cubo (o hexaedro regular), icosaedro y dodecaedro, y la indagaci´on de alguna de sus propiedades. As´ı, establece que la arista del icosaedro es la cantidad irracional llamada menor, mientras que la del dodecaedro es la cantidad irracional denominada ap´otoma.

In this Final Degree Project we study the last three books of Euclid’s Elements, which deal with the geometry in the space. At the beginning of Book XI we can find the 28 definitions employed in these books: solids, stright line or plane at right angles to a plane, pyramid, prism, sphere, cone, cylinder and the five regular polyhedron, amongst others. The greatest part of Book XI, with 39 propositions, focalizes in a detailed analysis of the parallelepiped solids. In Book XII, trough 18 propositions, Euclid connects the volumes of prisms and pyramids with those ones of cylinders and cones, respectively, as well as the volumes of spheres with the cubed ratio of their corresponding diameters, by means of a masterful use of the theory of proportion and the method of exhaustion. Finally, the main objetive of Book XIII and a very satisfactory finish to this magnum opus of Euclid, is the construction of the regular polyhedrons (platonic solids): tetrahedron, cube, octahedron, dodecahedron and icosahedron, and the check of some of their proporties. In this way, it is established that the edge of icosahedron is the irrational straight-line called minor, while the edge of dodecahedron is the irrational one named apotome.