Download Geometry by its history by Alexander Ostermann, Gerhard Wanner (auth.) PDF
By Alexander Ostermann, Gerhard Wanner (auth.)
In this textbook the authors current first-year geometry approximately within the order during which it was once stumbled on. the 1st 5 chapters express how the traditional Greeks demonstrated geometry, including its quite a few useful purposes, whereas more moderen findings on Euclidian geometry are mentioned in addition. the next 3 chapters clarify the revolution in geometry as a result growth made within the box of algebra via Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are taken care of in chapters nine and 10. The final bankruptcy bargains an advent to projective geometry, which emerged within the 19thcentury.
Complemented by way of quite a few examples, workouts, figures and images, the publication bargains either motivation and insightful causes, and gives stimulating and relaxing interpreting for college students and lecturers alike.
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24, right), Fig. 24. Cone and cylinder a cube (κ˘υβος; see Fig. 25, left), an octahedron (ὀκτάεδρον from ὀκτάεδρος – eight-sided; see Fig. 25, right) Fig. 25. Cube and octahedron an icosahedron (εἰκοσάεδρον; see Fig. 26, left), and finally a dodecahedron (δωδεκάεδρον; see Fig. 26, right). Fig. 26. Icosahedron and dodecahedron The four last ones, together with the tetrahedron (τετράεδρον, with four faces) which Euclid does not define, form the class of regular polyhedra. This class is identical to that of the Platonic solids or cosmic figures; Plato described them in his Timæus and associated them to the five elements (cube ↔ earth, icosahedron ↔ water, octahedron ↔ air, tetrahedron ↔ fire, dodecahedron ↔ ether).
1 and 2) and compass (Post. 3) to an intellectual level. The fourth postulate expresses the homogeneity of space in all directions by using the right angle as a universal measure for angles; the fifth postulate, finally, is the celebrated parallel postulate. Over the centuries, it gave rise to many discussions. The postulates are followed by common notions (also called axioms in some translations) which comprise the usual rules for equations and inequalities. The propositions. Then starts the sequence of propositions which develops the entire geometry from the definitions, the five postulates, the axioms and from propositions already proved.
30. Fig. 28. Self-duality of tetrahedron Fig. 29. Duality between cube and octahedron Fig. 30. Duality between icosahedron and dodecahedron 48 2 The Elements of Euclid Euclid omitted the definition of the parallelepiped (παραλληλεπίπεδον, a solid with parallel surfaces) and of the right-angled parallelepiped (where all angles are right), see Fig. 31. Fig. 31. Parallelepiped and right-angled parallelepiped Eucl. 26. Properties of planes, lines and angles in space. We postpone these questions to Part II where we will discuss them using tools from linear algebra.