Elementary Mathematics from a Higher Standpoint: Volume III: Precision Mathematics and Approximation Mathematics

✍ Scribed by Klein, Felix;Menghini, Marta(Translation)

• Publisher: Springer
• Year: 2016
• Tongue: English
• Leaves: 267
• Edition: 3
• Category: Library

✦ Synopsis

These three volumes constitute the first complete English translation of Felix Klein s seminal series Elementarmathematik vom hoheren 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.

In Volume III, Klein explores the relationship between precision and approximation mathematics. He crosses the various fields of mathematics from functions in one and two variables to practical geometry to space curves and surfaces underlining the relation between the exactness of the idealised concepts and the approximations to be considered in applications. Logical procedures are confronted with the way in which concepts arise starting from observations. It is a comparison between properties pertaining only to the theoretical field of abstract mathematics and properties that can be grasped by intuition. The final part, which concerns "gestalt" relations of curves and surfaces, shows Klein to be the master of the art of description of geometric forms."

✦ Table of Contents

Preface to the 2016 Edition......Page 5
Preface to the First Edition......Page 9
Preface to the Second Edition......Page 11
Preface to the Third Edition......Page 12
Contents......Page 14
Introduction......Page 18
First Part: On Functions of a Real Variable and Their Representation in an Orthogonal Coordinate System......Page 20
Empirical and Abstract Precision. The Modern Concept of Number......Page 21
Precision and Approximation Mathematics, also in Pure Geometry......Page 23
Intuition and Thought, Explained by Different Examples from Geometry......Page 26
Explanation Through Two Simple Propositions on Point Sets......Page 28
The Abstract and the Empirical Definition of a Function (Idea of the Function Stripe)......Page 31
On the Efficiency of Space Intuition......Page 34
On the Exactness of the Laws of Nature (with a Digression About Differing Conceptions Regarding the Constitution of Matter)......Page 35
Properties of the Empirical Curve: Connectedness, Slope, Curvature......Page 38
Cauchy's Definition of the Continuous Curve. How Far-Reaching is the Analogy with an Empirical Curve?......Page 43
The Integrability of Continuous Functions......Page 47
The Theorem of the Existence of the Greatest and of the Smallest Value......Page 51
The Four Derivates......Page 53
Weierstraß's Non-Differentiable Function. Its General Characteristics......Page 57
Its Non-Differentiability......Page 63
The “Reasonable” Functions......Page 69
Approximation of Empirical Curves by Means of Reasonable Functions......Page 70
Approximation of a Reasonable Function by Means of Simple Analytic Expressions......Page 72
Lagrange's Interpolation Formula......Page 73
Taylor's Theorem and Taylor's Series......Page 74
Approximation of the Integral and of the Derivative Using Lagrange's Polynomial......Page 78
Analytic Functions and Their Use in the Explanation of Nature......Page 81
Interpolation by Means of Trigonometric Series......Page 86
Evaluation of the Error in the Representation of Empirical Functions......Page 90
Trigonometric Interpolation According to the Least Squares Method......Page 93
The Harmonic Analyser......Page 94
Examples of Trigonometric Series......Page 97
Chebyshev's Works on Interpolation......Page 102
Continuity......Page 105
Interchangeability of the Order of Differentiation.......Page 111
Approximate Representation of Functions of the Spherical Surface by Means of Series of Spherical Harmonics......Page 117
Distribution of the Values of a Spherical Function over the Sphere......Page 124
Estimate of the Error Truncating the Series of Spherical Harmonics......Page 126
Second Part: Free Geometry of Plane Curves......Page 128
Theorems About Point Sets......Page 129
Point Sets Obtained by Inversion with Respect to Two or More Disjoint Circles......Page 130
Properties of These Point Sets......Page 136
The Concept of Bi-Dimensional Continuum. The General Concept of Curve......Page 138
About the Peano-Curve that Fills a Whole Square......Page 140
A More Specific Concept of Curve: The Jordan Curve......Page 147
Other Limitations to the Concept of Curve: The Regular Curve......Page 151
Approximation of intuitive Curves by Means of Regular Functions......Page 152
Perception of Idealised Curves......Page 153
Classification of Idealised Curves: Analytical and Algebraic Curves. Geometrical Construction of the Latter as Proposed by Graßmann......Page 154
Mastery of the Empirical Phenomena by Means of Idealised Structures: Perry's Point of View......Page 158
Iterated Inversion with Respect to Two Touching Circles......Page 160
The Same with Respect to Three Touching Circles (Modular Figure)......Page 164
The Normal Case of Four Circles Touching Each Other in a Cyclic Succession......Page 169
The General Case of Four Circles Touching Each Other in a Cyclic Succession......Page 171
Properties of the Non-Analytic Curves thus Arising......Page 175
Premises to the Whole Development. Further Idealization by Veronese......Page 180
Inaccuracy of All Practical Measurements. Examples from the Snellius Problem......Page 182
Determination of the Precision's Degree by Repeated Measurements. Interpretation in Principle of the Method of Least Squares......Page 185
Approximated Calculations, Explained by Means of Legendre's Theorem for Small Spherical Triangles......Page 186
The Geodetic Meaning of Shortest Line on the Earth-Spheroid (with Postulates Concerning the Theory of Differential Equations)......Page 188
About the Geoid and its Practical Determination......Page 191
Postulation of a Theory of Errors for Drawing Geometry, Explained Using a Graphic Reproduction of Pascal's Theorem......Page 195
About the Possibility to Deduce Properties of the Idealised Curve from the Empirical Shape......Page 199
Application of the Procedure on Algebraic Curves in Particular. Prerequisites Assumed in Algebra......Page 201
Establishing the Theorem We Want to Prove: w + 2 t = n (n - 2)......Page 206
Principles of the Continuity Proof to be Presented......Page 208
Transition of a C_n to a Shape with a Double Point......Page 211
Examples of Curves for Which the Theorem Holds; Case of Even n......Page 214
The Same for Odd n......Page 219
Explanation of the Continuity Proof Referring to Examples. Performing the Proof......Page 222
Third Part: About the Perception of Idealised Structures by Means of Drawings and Models......Page 229
Gestalt Relations of Non-Singular Space Curves, in Particular C_3 (Projections of the Curve and Plane Sections of its Tangent Surface)......Page 230
The Seven Kinds of Singular Points of Space Curves......Page 239
General Considerations on the Shape of Non-Singular Surfaces......Page 242
About the Double Points of the F_3, in Particular its Biplanar and Uniplanar Points......Page 245
About the General Behaviour of the F_3......Page 252
Appeal for an Always Renewed Adjustment of the Traditional Operating Mode of Science by Means of the Observation of Nature......Page 258
Name Index......Page 260
Subject Index......Page 263