Mathematics for the Nonmathematician

Morris Kline

Courier Dover Publications | 1985 | 658 páginas | djvu | 11 Mb

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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 problems.

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.

Mathematics and the Physical World

Morris Kline

Dover Publications | 1981 | 496 páginas | djvu | 3,6 Mb

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Stimulating account of development of basic mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations and non-Euclidean geometries. Also describes how math is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, other phenomena.

Mathematics : A Cultural Approach

Morris Kline

Addison-Wesley | 1963 | 701 páginas

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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

Mathematics in Aristotle

Thomas Heath

Clarendon Press | 1949 | 303 páginas | djvu | 2,8 Mb

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Mathematics in Aristotle is the result of work done during the last years of my husband's life, which he devoted to reading all that had been written on the subject and to making his own translations from the Greek.

His eagerness to return to this work too soon after a serious illness in 1939 was probably instrumental in hastening his end.

After his death in 1940 I found the manuscript (which I took to be unfinished) and consulted Sir David Ross as to the possibility of getting it completed. He most kindly read it through and, to my immense satisfaction, reported that the rough copy was, in fact, complete and advised that the fair copy, much of which was already done, should be finished in typescript.

This I undertook to do myself, and it proved not only a welcome distraction from the more disturbing elements of' total war', but, in addition, an unexpectedly enjoyable introduction to the fascinating world of Aristotle, which I had hitherto regarded as far above the head of one not a classical scholar. I had, however, not failed to remark the concomitance of Aristotelian learning with a love of music. During a short visit to Corpus Christi College during the presidency of Thomas Case, I had the unforgettable experience, having returned somewhat early from a Balliol Sunday concert, of discovering the President and my husband playing, by candle-light on the Broadwood grand piano, with unrestrained fortissimo, the four-hand arrangements of Haydn Symphonies. The conventional fiction obtained that my being a professional pianist precluded such indulgence within my hearing!

My very grateful thanks are due to Sir David Ross for his invaluable help and encouragement. For the reading of the proofs I am indebted to him and to Mr. Ivor Thomas, M.P., whose knowledge of Greek mathematics has been of the greatest assistance. Having no Greek myself, I have to thank Mr. Kerry Downes and Mr. P. R. W. Holmes for their assistance in transcribing the Greek quotations, and also my son Geoffrey T. Heath who, since his demobilization, has helped me in reading proofs and revising Greek quotations, mathematical figures, and formulae.

I regret the absence of a bibliography, as my husband left no note of authorities consulted, though some indications are given in footnotes as well as in the text.

The Copernicus of Antiquity - Aristarchus of Samos

Thomas L. Heath

online: archive.org

Apollonius of Perga Treatise on Conic Sections

Thomas L. Heath

Cambridge, University press | 1896

online: archive.org

This early work on conic sections is both expensive and hard to find in its first edition. It contains a comprehensive account of the mathematics of conics as expressed by Apollonius of Perga. This is a fascinating work and highly recommended for anyone interested in learning about mathematics and its history.

Índice

Introduction: pt. I. The earlier history of conic sections among the Greeks. 1. The discovery of conic sections; Menaechmus. 2. Aristaeus and Euclid. 3. Archimedes. pt. II. Introduction to the conics of Apollonius. 1. The author and his own account of the conics. 2. General characteristics. 3. The methods of Apollonius. 4. The construction of a conic by means of tangents. 5. The three-line and four-line locus. 6. The construction of a conic through five points.-Appendix: Notes on the terminology of Greek geometry.--The conics of Apollonius

The Works of Archimedes

Thomas L. Heath

PDF| 18,45 MB
on-line: archive.org

Diophantus of Alexandria: A Study in the History of Greek Algebra

Second Edition.
Thomas L. Heath
Cambridge, University Press | 1910| 387 páginas | PDF| 39,47 MB

online: archive.org