Erudite and entertaining overview follows development of mathematics from ancient Greeks to present. Topics include logic and mathematics, the fundamental concept, differential calculus, probability theory, much more. Exercises and problems.
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