Mathematics for the Non-mathematician

Morris Kline: Mathematics for the MassesMorris 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

• 1 Why Mathematics?2 A Historical Orientation2-1 Introduction2-2 Mathematics in early civilizations2-3 The classical Greek period2-4 The Alexandrian Greek period2-5 The Hindus and Arabs2-6 Early and medieval Europe2-7 The Renaissance2-8 Developments from 1550 to 18002-9 Developments from 1800 to the present2-10 The human aspect of mathematics3 Logic and Mathematics3-1 Introduction3-2 The concepts of mathematics3-3 Idealization3-4 Methods of reasoning3-5 Mathematical proof3-6 Axioms and definitions3-7 The creation of mathematics4 Number: the Fundamental Concept4-1 Introduction4-2 Whole numbers and fractions4-3 Irrational numbers4-4 Negative numbers4-5 The axioms concerning numbers* 4-6 Applications of the number system5 "Algebra, the Higher Arithmetic"5-1 Introduction5-2 The language of algebra5-3 Exponents5-4 Algebraic transformations5-5 Equations involving unknowns5-6 The general second-degree equation* 5-7 The history of equations of higher degree6 The Nature and Uses of Euclidean Geometry6-1 The beginnings of geometry6-2 The content of Euclidean geometry6-3 Some mundane uses of Euclidean geometry* 6-4 Euclidean geometry and the study of light6-5 Conic sections* 6-6 Conic sections and light* 6-7 The cultural influence of Euclidean geometry7 Charting the Earth and Heavens7-1 The Alexandrian world7-2 Basic concepts of trigonometry7-3 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 Nature8-1 The Greek concept of nature8-2 Pre-Greek and Greek views of nature8-3 Greek astronomical theories8-4 The evidence for the mathematical design of nature8-5 The destruction of the Greek world* 9 The Awakening of Europe9-1 The medieval civilization of Europe9-2 Mathematics in the medieval period9-3 Revolutionary influences in Europe9-4 New doctrines of the Renaissance9-5 The religious motivation in the study of nature* 10 Mathematics and Painting in the Renaissance10-1 Introduction10-2 Gropings toward a scientific system of perspective10-3 Realism leads to mathematics10-4 The basic idea of mathematical perspective10-5 Some mathematical theorems on perspective drawing10-6 Renaissance paintings employing mathematical perspective10-7 Other values of mathematical perspective11 Projective Geometry11-1 The problem suggested by projection and section11-2 The work of Desargues11-3 The work of Pascal11-4 The principle of duality11-5 The relationship between projective and Euclidean geometries12 Coordinate Geometry12-1 Descartes and Fermat12-2 The need for new methods in geometry12-3 The concepts of equation and curve12-4 The parabola12-5 Finding a curve from its equation12-6 The ellipse* 12-7 The equations of surfaces* 12-8 Four-dimensional geometry12-9 Summary13 The Simplest Formulas in Action13-1 Mastery of nature13-2 The search for scientific method13-3 The scientific method of Galileo13-4 Functions and formulas13-5 The formulas describing the motion of dropped objects13-6 The formulas describing the motion of objects thrown downward13-7 Formulas for the motion of bodies projected upward14 Parametric Equations and Curvillinear Motion14-1 Introduction14-2 The concept of parametric equations14-3 The motion of a projectile dropped from an airplane14-4 The motion of projectiles launched by cannons* 14-5 The motion of projectiles fired at an arbitrary angle14-6 Summary15 The Application of Formulas to Gravitation15-1 The revolution in astronomy15-2 The objections to a heliocentric theory15-3 The arguments for the heliocentric theory15-4 The problem of relating earthly and heavenly motions15-5 A sketch of Newton's life15-6 Newton's key idea15-7 Mass and weight15-8 The law of gravitation15-9 Further discussion of mass and weight15-10 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 Calculus16-1 Introduction16-2 The problem leading to the calculus16-3 The concept of instantaneous rate of change16-4 The concept of instantaneous speed16-5 The method of increments16-6 The method of increments applied to general functions16-7 The geometrical meaning of the derivative16-8 The maximum and minimum values of functions* 17 The Integral Calculus17-1 Differential and integral calculus compared17-2 Finding the formula from the given rate of change17-3 Applications to problems of motion17-4 Areas obtained by integration17-5 The calculation of work17-6 The calculation of escape velocity17-7 The integral as the limit of a sum17-8 Some relevant history of the limit concept17-9 The Age of Reason18 Trigonometric Functions and Oscillatory Motion18-1 Introduction18-2 The motion of a bob on a spring18-3 The sinusoidal functions18-4 Acceleration in sinusoidal motion18-5 The mathematical analysis of the motion of the bob18-6 Summary* 19 The Trigonometric Analysis of Musical Sounds19-1 Introduction19-2 The nature of simple sounds19-3 The method of addition of ordinates19-4 The analysis of complex sounds19-5 Subjective properties of musical sounds20 Non-Euclidean Geometries and Their Significance20-1 Introduction20-2 The historical background20-3 The mathematical content of Gauss's non-Euclidean geometry20-4 Riemann's non-Euclidean geometry20-5 The applicability of non-Euclidean geometry20-6 The applicability of non-Euclidean geometry under a new interpretation of line20-7 Non-Euclidean geometry and the nature of mathematics20-8 The implications of non-Euclidean geometry for other branches of our culture21 Arithmetics and Their Algebras21-1 Introduction21-2 The applicability of the real number system21-3 Baseball arithmetic21-4 Modular arithmetics and their algebras21-5 The algebra of sets21-6 Mathematics and models* 22 The Statistical Approach to the Social and Biological Sciences22-1 Introduction22-2 A brief historical review22-3 Averages22-4 Dispersion22-5 The graph and normal curve22-6 Fitting a formula to data22-7 Correlation22-8 Cautions concerning the uses of statistics* 23 The Theory of Probability23-1 Introduction23-2 Probability for equally likely outcomes23-3 Probability as relative frequency23-4 Probability in continuous variation23-5 Binomial distributions23-6 The problems of sampling24 The Nature and Values of Mathem24-4 The aesthetic and intellectual values24-5 Mathematics and rationalism24-6 The limitations of mathematicsTable of Trigonometric RatiosAnswers to Selected and Review ExercisesAdditional Answers and SolutionsIndex