Elementary Mathematics from a Higher Standpoint: Volume II: Geometry

Description

These three volumes constitute the first complete English translation of Felix Klein’s seminal series “Elementarmathematik vom höheren Standpunkte aus”. “Complete” has a twofold meaning here: First, there now exists a translation of volume III into English, while until today the only translation had been into Chinese. Second, the English versions of volume I and II had omitted several, even extended parts of the original, while we now present a complete revised translation into modern English. The volumes, first published between 1902 and 1908, are lecture notes of courses that Klein offered to future mathematics teachers, realizing a new form of teacher training that remained valid and effective until today: Klein leads the students to gain a more comprehensive and methodological point of view on school mathematics. The volumes enable us to understand Klein’s far-reaching conception of elementarisation, of the “elementary from a higher standpoint”, in its implementation for school mathematics. This volume II presents a paradigmatic realisation of Klein’s approach of elementarisation for teacher education. It is shown how the various geometries, elaborated particularly since the beginning of the 19th century, are revealed as becoming unified in a new restructured geometry. As Klein liked to stress: “Projective geometry is all geometry”. Non-Euclidean geometry proves to constitute a part of this unifying process. The teaching of geometry is discussed in a separate chapter, which provides moreover important information on the history of geometry teaching and an international comparison. Front Matter....Pages i-xvi Introduction....Pages 1-2 Front Matter....Pages 3-3 I. Line-segment, Area, Volume as Relative Quantities....Pages 5-26 II. The Grassmannian Determinant Principle for the Plane....Pages 27-35 III. The Grassmannian Principle for Space....Pages 37-48 IV. Classification of the Elementary Configuration of Space Accoring to Their Behaviour Under Transformation of Rectangular Coordinates....Pages 49-64 V. Higher Configurations....Pages 65-79 Front Matter....Pages 81-82 I. Affine Transformations....Pages 83-100 II. Projective Transformations....Pages 101-113 III. Higher Point Transformations....Pages 115-126 IV. Tansformations with Change of Space Element....Pages 127-136 V. Theory of the Imaginary....Pages 137-150 Front Matter....Pages 151-151 I. The Systematic Discussion....Pages 153-182 II. Foundations of Geometry....Pages 183-235 Front Matter....Pages 237-241 I. The Teaching in England....Pages 243-247 II. The Teaching in France....Pages 249-257 III. The Teaching in Italy....Pages 259-263 IV. The Teaching in Germany....Pages 265-276 Back Matter....Pages 277-315