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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. Why Mathematics? A Historical Orientation Introduction Mathematics in early civilizations The classical Greek period The Alexandrian Greek period The Hindus and Arabs Early and medieval Europe The Renaissance Developments from 1550 to 1800 Developments from 1800 to the present The human aspect of mathematics Logic and Mathematics Introduction The concepts of mathematics Idealization Methods of reasoning Mathematical proof Axioms and definitions The creation of mathematics Number: the Fundamental Concept Introduction Whole numbers and fractions Irrational numbers Negative numbers The axioms concerning numbers Applications of the number system Algebra, the Higher Arithmetic Introduction The language of algebra Exponents Algebraic transformations Equations involving unknowns The general second-degree equation The history of equations of higher degree The Nature and Uses of Euclidean Geometry The beginnings of geometry The content of Euclidean geometry Some mundane uses of Euclidean geometry Euclidean geometry and the study of light Conicsections Conic sections and light The cultural influence of Euclidean geometry Charting the Earth and the Heavens The Alexandrian world Basic concepts of trigonometry Some mundane uses of trigonometric ratios Charting the earth Charting the heavens Further progress in the study of light The Mathematical Order of Nature The Greek concept of nature Pre-Greek and Greek views of nature Greek astronomical theories The evidence for the mathematical design of nature The destruction of the Greek world The Awakening of Europe The medieval civilization of Europe Mathematics in the medieval period Revolutionary influences in Europe New doctrines of the Renaissance The religious motivation in the study of nature Mathematics and Painting in the Renaissance Introduction Gropings toward a scientific system of perspective Realism leads to mathematics The basic idea of mathematical perspective Some mathematical theorems on perspective drawing Renaissance paintings employing mathematical perspective Other values of mathematical perspective Projective Geometry The problem suggested by projection and section The work of Desargus The work of Pascal The principle of duality The relationship between projective and Euclidean geometries 247 Coordinate Geometry Descartes and Fermat The need for new methods in geometry The concepts of equation and curve The parabola Finding a curve from its equation The ellipse The equations of surfaces Four-dimensional geometry Summary The Simplest Formulas in Action Mastery of nature The search for scientific method The scientific method of Galileo Functions and formulas The formulas describing the motion of dropped objects The formulas describing the motion of objects thrown downward Formulas for the motion of bodies projected upward Parametric Equations and Curvilinear Motion Introduction The concept of parametric equations The motion of a projectile dropped from an airplane The motion of projectiles launched by cannons The motion of projectiles fired at an arbitrary angle Summary The Application of Formulas to Gravitation The revolution in astronomy The objections to a heliocentric theory The arguments for the heliocentric theory The problem of relating earthly and heavenly motions A sketch of Newton’s life Newton’s key idea Mass and weight The law of gravitation Further discussion of mass and weight Some deductions from the law of gravitation The rotation of the earth Gravitation and the Keplerian laws Implications of the theory of gravitation The Differential Calculus Introduction The problems leading to the calculus The concept of instantaneous rate of change The concept of instantaneous speed The method of increments The method of increments applied to general functions The geometrical meaning of the derivative The maximum and minimum values of functions The Integral Calculus Differential and integral calculus compared Finding the formula from the given rate of change Applications to problems of motion Areas obtained by integration The calculation of work The calculation of escape velocity The integral as the limit of a sum Some relevant history of the limit concept The Age of Reason Trigonometric Functions and Oscillatory Motion Introduction The motion of a bob on a spring The sinusoidal fu nctions Acceleration in sinusoidal motion The mathematical analysis of the motion of the bob Summary The Trigonometric Analysis of Musical Sounds Introduction The nature of simple sounds The method of addition of ordinates The analysis of complex sounds Subjective properties of musical sounds Non-Euclidean Geometries and Their Significance Introduction The historical background The mathematical content of Gauss’s non-Euclidean geometry Riemann’s non-Euclidean geometry The applicability of non-Euclidean geometry The applicability of non-Euclidean geometry under a new interpretation of line Non-Euclidean geometry and the nature of mathematics The implications of non-Euclidean geometry for other branches of our culture Arithmetics and Their Algebras Introduction The applicability of the real number system Baseball arithmetic Modular arithmetics and their algeras The algebra of sets Mathematics and models The Statistical Approach to the Social and Biological Sciences Introduction A brief historical review Averages Dispersion The graph and the normal curve Fitting a formula to data Correlation Cautions concerning the uses of statistics The Theory of Probability Introduction Probability for equally likely outcomes Probability as relative frequency Probability in continuous variation Binomial distributions The problems of sampling The Nature and Values of Mathematics Introduction The structure of mathematics The values of mathematics for the study of nature The aesthetic and intellectual values Mathematics and rationalism The limitations of mathematics Table of Trigonometric Ratios Answers to Selected and Review Exercises Additional Answers and Solutions Index