Kline Morris. Mathematics for the Nonmathematician

Courier Corporation, 1985. — 641 p. — (Dover books explaining science). — ISBN: 9780486248233, 0486248232.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 OrientationMathematics 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 MathematicsThe concepts of mathematics
Idealization
Methods of reasoning
Mathematical proof
Axioms and definitions
The creation of mathematics
Number: the Fundamental ConceptWhole numbers and fractions
Irrational numbers
Negative numbers
The axioms concerning numbers
Applications of the number system
Algebra, the Higher ArithmeticThe 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 RenaissanceGropings 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 geometryThe 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 MotionThe 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 angleThe 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 CalculusThe 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 MotionThe motion of a bob on a spring
The sinusoidal fu nctions
Acceleration in sinusoidal motion
The mathematical analysis of the motion of the bobThe Trigonometric Analysis of Musical SoundsThe 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 SignificanceThe 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 AlgebrasThe 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 SciencesA 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 ProbabilityProbability for equally likely outcomes
Probability as relative frequency
Probability in continuous variation
Binomial distributions
The problems of sampling
The Nature and Values of MathematicsThe 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