Table of Contents

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 mathematics 3 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 mathematics 4 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 system 5 "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 degree 6 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 geometry 7 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 light 8 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 perspective 11 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 geometries 12 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 Summary 13 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 upward 14 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 Summary 15 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 Reason 18 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 sounds 20 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 culture 21 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 sampling 24 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

About the Author

Morris Kline: Mathematics for the Masses Morris Kline (1908-1992) had a strong and forceful personality which he brought both to his position as Professor at New York University from 1952 until his retirement in 1975, and to his role as the driving force behind Dover's mathematics reprint program for even longer, from the 1950s until just a few years before his death. Professor Kline was the main reviewer of books in mathematics during those years, filling many file drawers with incisive, perceptive, and always handwritten comments and recommendations, pro or con. It was inevitable that he would imbue the Dover math program - which he did so much to launch - with his personal point of view that what mattered most was the quality of the books that were selected for reprinting and the point of view that stressed the importance of applications and the usefulness of mathematics. He urged that books should concentrate on demonstrating how mathematics could be used to solve problems in the real world, not solely for the creation of intellectual structures of theoretical interest to mathematicians only. Morris Kline was the author or editor of more than a dozen books, including Mathematics in Western Culture (Oxford, 1953), Mathematics: The Loss of Certainty (Oxford, 1980), and Mathematics and the Search for Knowledge (Oxford, 1985). His Calculus, An Intuitive and Physical Approach, first published in 1967 and reprinted by Dover in 1998, remains a widely used text, especially by readers interested in taking on the sometimes daunting task of studying the subject on their own. His 1985 Dover book, Mathematics for the Nonmathematician could reasonably be regarded as the ultimate math for liberal arts text and may have reached more readers over its long life than any other similarly directed text. In the Author's Own Words: "Mathematics is the key to understanding and mastering our physical, social and biological worlds." "Logic is the art of going wrong with confidence." "Statistics: the mathematical theory of ignorance." "A proof tells us where to concentrate our doubts." - Morris Kline