Mathematics for the Nonmathematician

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