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.
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.
Erudite and entertaining overview follows development of mathematics from ancient Greeks, through Middle Ages and Renaissance to the present. Chapters focus on Logic and Mathematics, the Number, the Fundamental Concept, Differential Calculus, the Theory of Probability and much more. Exercises and pr
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