Kline Morris. Mathematics for liberal arts

Reading, MA: Addison-Wesley Publishing Company, 1967. — 654 p.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 to
Developments from to the present.
The human aspect of mathematicsLogic and MathematicsThe concepts of mathematics.
Idealization.
Methods of reasoning.
Mathematical proof
Axioms and definitions.
The creation of mathematicsNumber: 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.
Conic sections.
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 Desargues.
The work of Pascal.
The principle of duality.
The relationship between projective and Euclidean geometriesCoordinate 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 upwardParametric 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 gravitationThe 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 functionsThe 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 functions
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 cultureArithmetics and Their AlgebrasThe applicability of the real number system
Baseball arithmetic
Modular arithmetics and their algebras
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 statisticsThe Theory of ProbabilityProbability for equally likely outcomes.
Probability as relative frequency
Probability in continuous variation.
Binomial distributions
The problems of samplingThe 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 mathematicsTable of Trigonometric Ratios
Answers to Selected and Review Exercises
Additional Answers and Solutions