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Mathematics for the Nonmathematician

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Morris Kline
Dover Publications
eBook Typ:
Adobe Digital Editions
eBook Format:
2 - DRM Adobe
1 Why Mathematics?2 A Historical Orientation 2-1 Introduction 2-2 Mathematics in early civilizations 2-3 The classical Greek period 2-4 The Alexandrian Greek period 2-5 The Hindus and Arabs 2-6 Early and medieval Europe 2-7 The Renaissance 2-8 Developments from 1550 to 1800 2-9 Developments from 1800 to the present 2-10 The human aspect of mathematics3 Logic and Mathematics 3-1 Introduction 3-2 The concepts of mathematics 3-3 Idealization 3-4 Methods of reasoning 3-5 Mathematical proof 3-6 Axioms and definitions 3-7 The creation of mathematics4 Number: the Fundamental Concept 4-1 Introduction 4-2 Whole numbers and fractions 4-3 Irrational numbers 4-4 Negative numbers 4-5 The axioms concerning numbers * 4-6 Applications of the number system5 "Algebra, the Higher Arithmetic" 5-1 Introduction 5-2 The language of algebra 5-3 Exponents 5-4 Algebraic transformations 5-5 Equations involving unknowns 5-6 The general second-degree equation * 5-7 The history of equations of higher degree6 The Nature and Uses of Euclidean Geometry 6-1 The beginnings of geometry 6-2 The content of Euclidean geometry 6-3 Some mundane uses of Euclidean geometry * 6-4 Euclidean geometry and the study of light 6-5 Conic sections * 6-6 Conic sections and light * 6-7 The cultural influence of Euclidean geometry7 Charting the Earth and Heavens 7-1 The Alexandrian world 7-2 Basic concepts of trigonometry 7-3 Some mundane uses of trigonometric ratios * 7-4 Charting the earth * 7-5 Charting the heavens * 7-6 Further progress in the study of light8 The Mathematical Order of Nature 8-1 The Greek concept of nature 8-2 Pre-Greek and Greek views of nature 8-3 Greek astronomical theories 8-4 The evidence for the mathematical design of nature 8-5 The destruction of the Greek world* 9 The Awakening of Europe 9-1 The medieval civilization of Europe 9-2 Mathematics in the medieval period 9-3 Revolutionary influences in Europe 9-4 New doctrines of the Renaissance 9-5 The religious motivation in the study of nature* 10 Mathematics and Painting in the Renaissance 10-1 Introduction 10-2 Gropings toward a scientific system of perspective 10-3 Realism leads to mathematics 10-4 The basic idea of mathematical perspective 10-5 Some mathematical theorems on perspective drawing 10-6 Renaissance paintings employing mathematical perspective 10-7 Other values of mathematical perspective11 Projective Geometry 11-1 The problem suggested by projection and section 11-2 The work of Desargues 11-3 The work of Pascal 11-4 The principle of duality 11-5 The relationship between projective and Euclidean geometries12 Coordinate Geometry 12-1 Descartes and Fermat 12-2 The need for new methods in geometry 12-3 The concepts of equation and curve 12-4 The parabola 12-5 Finding a curve from its equation 12-6 The ellipse * 12-7 The equations of surfaces * 12-8 Four-dimensional geometry 12-9 Summary13 The Simplest Formulas in Action 13-1 Mastery of nature 13-2 The search for scientific method 13-3 The scientific method of Galileo 13-4 Functions and formulas 13-5 The formulas describing the motion of dropped objects 13-6 The formulas describing the motion of objects thrown downward 13-7 Formulas for the motion of bodies projected upward14 Parametric Equations and Curvillinear Motion 14-1 Introduction 14-2 The concept of parametric equations 14-3 The motion of a projectile dropped from an airplane 14-4 The motion of projectiles launched by cannons * 14-5 The motion of projectiles fired at an arbitrary angle 14-6 Summary15 The Application of Formulas to Gravitation 15-1 The revolution in astronomy 15-2 The objections to a heliocentric theory 15-3 The arguments for the heliocentric theory 15-4 The problem of relating earthly and heavenly motions 15-5 A sketch of Newton's life 15-6 Newton's key idea 15-7 Mass and weight 15-8 The law of gravitation 15-9 Further discussion of mass and weight 15-10 Some deductions from the law of gravitation * 15-11 The rotation of the earth * 15-12 Gravitation and the Keplerian laws * 15-13 Implications of the theory of gravitation* 16 The Differential Calculus 16-1 Introduction 16-2 The problem leading to the calculus 16-3 The concept of instantaneous rate of change 16-4 The concept of instantaneous speed 16-5 The method of increments 16-6 The method of increments applied to general functions 16-7 The geometrical meaning of the derivative 16-8 The maximum and minimum values of functions* 17 The Integral Calculus 17-1 Differential and integral calculus compared 17-2 Finding the formula from the given rate of change 17-3 Applications to problems of motion 17-4 Areas obtained by integration 17-5 The calculation of work 17-6 The calculation of escape velocity 17-7 The integral as the limit of a sum 17-8 Some relevant history of the limit concept 17-9 The Age of Reason18 Trigonometric Functions and Oscillatory Motion 18-1 Introduction 18-2 The motion of a bob on a spring 18-3 The sinusoidal functions 18-4 Acceleration in sinusoidal motion 18-5 The mathematical analysis of the motion of the bob 18-6 Summary* 19 The Trigonometric Analysis of Musical Sounds 19-1 Introduction 19-2 The nature of simple sounds 19-3 The method of addition of ordinates 19-4 The analysis of complex sounds 19-5 Subjective properties of musical sounds20 Non-Euclidean Geometries and Their Significance 20-1 Introduction 20-2 The historical background 20-3 The mathematical content of Gauss's non-Euclidean geometry 20-4 Riemann's non-Euclidean geometry 20-5 The applicability of non-Euclidean geometry 20-6 The applicability of non-Euclidean geometry under a new interpretation of line 20-7 Non-Euclidean geometry and the nature of mathematics 20-8 The implications of non-Euclidean geometry for other branches of our culture21 Arithmetics and Their Algebras 21-1 Introduction 21-2 The applicability of the real number system 21-3 Baseball arithmetic 21-4 Modular arithmetics and their algebras 21-5 The algebra of sets 21-6 Mathematics and models* 22 The Statistical Approach to the Social and Biological Sciences 22-1 Introduction 22-2 A brief historical review 22-3 Averages 22-4 Dispersion 22-5 The graph and normal curve 22-6 Fitting a formula to data 22-7 Correlation 22-8 Cautions concerning the uses of statistics* 23 The Theory of Probability 23-1 Introduction 23-2 Probability for equally likely outcomes 23-3 Probability as relative frequency 23-4 Probability in continuous variation 23-5 Binomial distributions 23-6 The problems of sampling24 The Nature and Values of Mathem 24-4 The aesthetic and intellectual values 24-5 Mathematics and rationalism 24-6 The limitations of mathematics Table of Trigonometric Ratios Answers to Selected and Review Exercises Additional Answers and Solutions Index
Practical, scientific, philosophical, and artistic problems have caused men to investigate mathematics. But there is one other motive which is as strong as any of these—the search for beauty. Mathematics is an art, and as such affords the pleasures which all the arts afford." In this erudite, entertaining college-level text, Morris Kline, Professor Emeritus of Mathematics at New York University, provides the liberal arts student with a detailed treatment of mathematics in a cultural and historical context. The book can also act as a self-study vehicle for advanced high school students and laymen. Professor Kline begins with an overview, tracing the development of mathematics to the ancient Greeks, and following its evolution through the Middle Ages and the Renaissance to the present day. Subsequent chapters focus on specific subject areas, such as "Logic and Mathematics," "Number: The Fundamental Concept," "Parametric Equations and Curvilinear Motion," "The Differential Calculus," and "The Theory of Probability." Each of these sections offers a step-by-step explanation of concepts and then tests the student's understanding with exercises and problems. At the same time, these concepts are linked to pure and applied science, engineering, philosophy, the social sciences or even the arts.
In one section, Professor Kline discusses non-Euclidean geometry, ranking it with evolution as one of the "two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century." His lucid treatment of this difficult subject starts in the 1800s with the pioneering work of Gauss, Lobachevsky, Bolyai and Riemann, and moves forward to the theory of relativity, explaining the mathematical, scientific and philosophical aspects of this pivotal breakthrough. Mathematics for the Nonmathematician exemplifies Morris Kline's rare ability to simplify complex subjects for the nonspecialist.