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CONTENTS
CHAPTER 1. WHY MATHEMATICS?
1
CHAPTER 2. THE COURSE OF MATHEMATICS: AN HISTORICAL ORIENTATION
11
2-1 Introduction
11
2-2 Mathematics in early civilizations
11
2-3 The classical Greek period
14
2-4 The Alexandrian Greek period
17
2-5 The Hindus and Arabs
18
2-6 Early and medieval Europe
19
2-7 The Renaissance
20
2-8 Developments from 1550 to 1800
21
2-9 Developments from 1800 to the present
24
2-10 The human aspect of mathematics
26
CHAPTER 3. THE WAYS OF MATHEMATICS
29
3-1 Introduction
29
3-2 The concepts of mathematics
29
3-3 Idealization
37
3-4 Methods of reasoning
38
3-5 Mathematical proof
41
3-6 Axioms and definitions
47
3-7 The creation of mathematics
48
3-8 Summary
52
CHAPTER 4. NUMBER: THE FUNDAMENTAL CONCEPT
54
4-1 Introduction
54
4-2 Whole numbers and fractions
54
4-3 Irrational numbers
59
4-4 Negative numbers
67
4-5 The axioms concerning numbers
69
4-6 Applications of the number system
72
CHAPTER 5. ALGEBRA, THE HIGHER ARITHMETIC
82
5-1 Introduction
82
5-2 The language of algebra
82
5-3 Algebraic transformations
85
5-4 Equations involving unknowns
88
CHAPTER 6. THE NATURE AND USES OF EUCLIDEAN GEOMETRY
103
6-1 The beginnings of geometry
103
6-2 The content of Euclidean geometry
105
6-3 Some mundane uses of Euclidean geometry
111
6-4 Euclidean geometry and the study of light
117
6-5 Conic sections
123
6-6 The conic sections and light
125
6-7 The cultural influence of Euclidean geometry
129
6-8 Euclidean geometry within the Greek cultural world
131
CHAPTER 7. CHARTING THE EARTH AND THE HEAVENS
136
7-1 The Alexandrian world
136
7-2 Basic concepts of trigonometry
141
7-3 Some mundane uses of trigonometric ratios
145
7-4 Charting the earth
148
7-5 Charting the heavens
154
7-6 Further progress in the study of light
158
CHAPTER 8. THE MATHEMATICAL ORDER OF NATURE
169
8-1 The Greek concept of nature
169
8-2 Pre-Greek and early Greek views of nature
169
8-3 The Pythagorean and Platonic views of nature
172
8-4 Greek astronomical theories
174
8-5 The evidence for the mathematical design of nature
177
8-6 The destruction of the Greek world
180
CHAPTER 9. THE AWAKENING OF EUROPE
183
9-1 The medieval civilization of Europe
183
9-2 Mathematics in the medieval period
186
9-3 First revival of the Greek works
189
9-4 Revolutionary influence in Europe
192
9-5 Intellectual revolt
195
9-6 New doctrines of the Renaissance
197
9-7 The religious motivation in the study of nature
200
CHAPTER 10. MATHEMATICS AND PAINTING IN THE RENAISSANCE
203
10-1 Introduction
203
10-2 Gropings toward a scientific system of perspective
203
10-3 Realism leads to mathematics
209
10-4 The basic idea of mathematical perspective
212
10-5 Some mathematical theorems on perspective drawing
215
10-6 Renaissance paintings employing mathematical perspective
221
10-7 Other values of mathematical perspective
229
CHAPTER 11. PROJECTIVE GEOMETRY
232
11-1 The problem suggested by projection and section
232
11-2 The work of Desargues
234
11-3 The work of Pascal
239
11-4 The principle of duality
242
11-5 The relationship between projective and Euclidean geometries
247
CHAPTER 12. THE REVOLUTION IN ASTRONOMY
250
12-1 Introduction
250
12-2 The work of Copernicus
250
12-3 The work of Kepler
253
12-4 The objections to a heliocentric theory
258
12-5 The arguments for the heliocentric theory
261
12-6 The scientific import of the heliocentric theory
264
12-7 The cultural influences of the heliocentric theory
265
CHAPTER 13. COORDINATE GEOMETRY
271
13-1 Descartes and Fermat
271
13-2 The need for new methods in geometry
274
13-3 The concepts of equation and curve
276
13-4 The parabola
284
13-5 Finding a curve from its equation
289
13-6 The ellipse
291
13-7 The equations of surfaces
293
13-8 Four-dimensional geometry
296
13-9 Summary
297
CHAPTER 14. THE MATHEMATIZATION OF SCIENCE
300
14-1 Mathematics and modern science
300
14-2 The growth of modern science
300
14-3 The search for scientific method
306
14-4 The scientific method of Galileo
308
CHAPTER 15. THE SIMPLEST FORMULAS IN ACTION
318
15-1 Introduction
318
15-2 Functions and formulas
318
15-3 The formulas describing the motion of dropped objects
321
15-4 The formulas describing the motion of objects thrown downward
327
15-5 Formulas for the motion of bodies projected upward
328
15-6 Mass and weight
331
15-7 Vertical motion in water
333
15-8 Motion along an inclined plane
335
15-9 Motion along planes with different slopes
341
15-10 Summary
344
CHAPTER 16. PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
346
16-1 Introduction
346
16-2 The concept of parametric equations
347
16-3 The motion of a projectile dropped from an airplane
348
16-4 The motion of projectiles launched by cannons
352
16-5 The motion of projectiles fired at an arbitrary angle
357
16-6 Summary
363
CHAPTER 17. THE APPLICATION OF FORMULAS TO GRAVITATION
365
17-1 Introduction
365
17-2 The problem of relating earthly and heavenly motions
365
17-3 A sketch of Newton's life
367
17-4 The law of gravitation and the second law of motion
369
17-5 Further discussion of weight and mass
375
17-6 Some deductions from the law of gravitation
377
17-7 The rotation of the earth
383
17-8 Gravitation and the Keplerian laws
386
17-9 Implications of the theory of gravitation
390
CHAPTER 18. THE DIFFERENTIAL CALCULUS
396
18-1 Introduction
396
18-2 The problems leading to the calculus
396
18-3 The concept of instantaneous rate of change
398
18-4 The concept of instantaneous speed
400
18-5 The method of increments
402
18-6 The method of increments applied to general functions
405
18-7 The geometrical meaning of the derivative
410
18-8 The maximum and minimum values of functions
413
CHAPTER 19. THE INTEGRAL CALCULUS
417
19-1 Differential and integral calculus compared
417
19-2 Finding the formula from the given rate of change
417
19-3 Applications to problems of motion
418
19-4 Areas obtained by integration
422
19-5 The calculation of work
426
19-6 The calculation of escape velocity
429
19-7 The equation of the cable of a bridge
431
19-8 The concept of a differential equation
432
19-9 The integral as the limit of a sum
434
19-10 Some relevant history of the limit concept
439
CHAPTER 20. THE AGE OF REASON
443
20-1 Introduction
443
20-2 Materialism
444
20-3 Mechanism
446
20-4 Determinism
448
20-5 Theories of knowledge
449
20-6 Psychology
455
20-7 Biology
457
20-8 Respite
459
CHAPTER 21. RELIGION IN THE AGE OF REASON
461
21-1 Introduction
461
21-2 The grounds of the conflict
461
21-3 The beliefs of the creators of the Age of Reason
463
21-4 Rational movements in religion
466
21-5 The decline of superstition
471
21-6 The rise of toleration
472
21-7 The problem of ethics
474
CHAPTER 22. REASON IN LITERATURE AND AESTHETICS
479
22-1 Introduction
479
22-2 The language of reason
479
22-3 The reform of style and spirit
481
22-4 The age of prose
485
22-5 The content of the literature
486
22-6 Aesthetics
487
22-7 The revolt against reason
489
22-8 Retrospect
491
CHAPTER 23. TRIGONOMETRIC FUNCTIONS AND OSCILLATORY MOTION
493
23-1 Introduction
493
23-2 The motion of a bob on a spring
494
23-3 The sinusoidal functions
495
23-4 The mathematical analysis of the motion of the bob
503
23-5 Summary
509
CHAPTER 24. THE TRIGONOMETRIC ANALYSIS OF MUSICAL SOUNDS
510
24-1 Introduction
510
24-2 The nature of simple sounds
511
24-3 The method of addition of ordinates
516
24-4 The analysis of complex sounds
518
24-5 Subjective properties of musical sounds
522
24-6 Some practical applications of the mathematical analysis
524
24-7 Applications to physiology
525
24-8 Summary
527
CHAPTER 25. TRIGONOMETRIC FUNCTIONS AND ELECTROMAGNETISM
529
25-1 Introduction
529
25-2 Historical background
529
25-3 The generation of alternating current
531
25-4 Electromagnetic waves
538
25-5 Electromagnetic waves and light
544
25-6 The range of electromagnetic waves
546
25-7 Electromagnetic waves and the sense of sight
546
25-8 Electromagnetic theory and the physical world
548
CHAPTER 26. NON-EUCLIDEAN GEOMETRIES AND THEIR SIGNIFICANCE
553
26-1 Introduction
553
26-2 The historical background
553
26-3 The mathematical content of Gauss's non-Euclidean geometry
560
26-4 Riemann's non-Euclidean geometry
562
26-5 The applicability of non-Euclidean geometry
563
26-6 The applicability of non-Euclidean geometry under a new interpreta- tion of line
566
26-7 Non-Euclidean geometry and the nature of mathematics
572
26-8 The implications of non-Euclidean geometry for other branches of our culture
575
CHAPTER 27. ARITHMETICS AND THEIR ALGEBRAS
579
27-1 Introduction
579
27-2 The applicability of the real number system
579
27-3 Modular arithmetics and their algebras
582
27-4 The algebra of sets
587
27-5 Arithmetics and algebras as structures
592
27-6 Mathematics and models
593
CHAPTER 28. THE DEDUCTIVE APPROACH TO THE SOCIAL SCIENCES
595
28-1 Introduction
595
28-2 The rational reconstruction of political science
595
28-3 The philosophy of utilitarianism
599
28-4 The rational approach to economics
603
28-5 The reform of the philosophy of history
608
28-6 The accomplishment in the social sciences
610
CHAPTER 29. THE STATISTICAL APPROACH TO THE SOCIAL AND BIOLOGICAL SCIENCES
613
29-1 Introduction
613
29-2 A brief historical review
613
29-3 Averages
615
29-4 Dispersion
617
29-5 The graph and the normal curve
619
29-6 Fitting a formula to data
625
29-7 Correlation
630
29-8 Cautions concerning the uses of statistics
632
CHAPTER 30. THE THEORY OF PROBABILITY
636
30-1 Introduction
636
30-2 Probability for equally likely outcomes
637
30-3 Probability as relative frequency
642
30-4 Probability in continuous variation
643
30-5 Binomial distributions
646
30-6 The problems of sampling
651
30-7 Probability in the physical sciences
652
30-8 The statistical view of nature
653
CHAPTER 31. THE NATURE AND VALUES OF MATHEMATICS
660
31-1 Introduction
660
31-2 The structure of mathematics
660
31-3 The values of mathematics for the study of nature
666
31-4 The aesthetic and intellectual values
669
31-5 Mathematics and rationalism
673
31-6 The limitations of mathematics
674
TABLE OF TRIGONOMETRIC RATIOS
679
ANSWERS TO SELECTED EXERCISES
683
INDEX
691
