Mathematics for the Nonmathematician

Contents......Page __sk_0009.djvu
1 Why Mathematics?......Page __sk_0015.djvu
2-2 Mathematics in early civilizations......Page __sk_0025.djvu
2-3 The classical Greek period......Page __sk_0028.djvu
2-4 The Alexandrian Greek period......Page __sk_0031.djvu
2-5 The Hindus and Arabs......Page __sk_0033.djvu
2-7 The Renaissance......Page __sk_0034.djvu
2-8 Developments from 1550 to 1800......Page __sk_0036.djvu
2-9 Developments from 1800 to the present......Page __sk_0038.djvu
2-10 The human aspect of mathematics......Page __sk_0041.djvu
3-2 The concepts of mathematics......Page __sk_0044.djvu
3-3 Idealization......Page __sk_0052.djvu
3-4 Methods of reasoning......Page __sk_0053.djvu
3-5 Mathematical proof......Page __sk_0059.djvu
3-6 Axioms and definitions......Page __sk_0064.djvu
3-7 The creation of mathematics......Page __sk_0065.djvu
4-2 Whole numbers and fractions......Page __sk_0072.djvu
4-3 Irrational numbers......Page __sk_0079.djvu
4-4 Negative numbers......Page __sk_0086.djvu
4-5 The axioms concerning numbers......Page __sk_0089.djvu
4-6 Applications of the number system......Page __sk_0096.djvu
5-2 The language of algebra......Page __sk_0108.djvu
5-3 Exponents......Page __sk_0111.djvu
5-4 Algebraic transformations......Page __sk_0116.djvu
5-5 Equations involving unknowns......Page __sk_0120.djvu
5-6 The general second-degree equation......Page __sk_0126.djvu
5-7 The history of equations of higher degree......Page __sk_0133.djvu
6-1 The beginnings of geometry......Page __sk_0137.djvu
6-2 The content of Euclidean geometry......Page __sk_0139.djvu
6-3 Some mundane uses of Euclidean geometry......Page __sk_0145.djvu
6-4 Euclidean geometry and the study of light......Page __sk_0152.djvu
6-5 Conic sections......Page __sk_0156.djvu
6-6 Conic sections and light......Page __sk_0158.djvu
6-7 The cultural influence of Euclidean geometry......Page __sk_0163.djvu
7-1 The Alexandrian world......Page __sk_0167.djvu
7-2 Basic concepts of trigonometry......Page __sk_0172.djvu
7-3 Some mundane uses of trigonometric ratios......Page __sk_0177.djvu
7-4 Charting the earth......Page __sk_0179.djvu
7-5 Charting the heavens......Page __sk_0185.djvu
7-6 Further progress in the study of light......Page __sk_0190.djvu
8-1 The Greek concept of nature......Page __sk_0201.djvu
8-2 Pre-Greek and Greek views of nature......Page __sk_0202.djvu
8-3 Greek astronomical theories......Page __sk_0204.djvu
8-4 The evidence for the mathematical design of nature......Page __sk_0206.djvu
8-5 The destruction of the Greek world......Page __sk_0208.djvu
9-1 The medieval civilization of Europe......Page __sk_0211.djvu
9-2 Mathematics in the medieval period......Page __sk_0213.djvu
9-3 Revolutionary influences in Europe......Page __sk_0214.djvu
9-4 New doctrines of the Renaissance......Page __sk_0216.djvu
9-5 The religious motivation in the study of nature......Page __sk_0220.djvu
10-1 Introduction......Page __sk_0223.djvu
10-2 Gropings toward a scientific system of perspective......Page __sk_0224.djvu
10-3 Realism leads to mathematics......Page __sk_0227.djvu
10-4 The basic idea of mathematical perspective......Page __sk_0229.djvu
10-5 Some mathematical theorems on perspective drawing......Page __sk_0233.djvu
10-6 Renaissance paintings employing mathematical perspective......Page __sk_0237.djvu
10-7 Other values of mathematical perspective......Page __sk_0243.djvu
11-1 The problem suggested by projection and section......Page __sk_0246.djvu
11-2 The work of Desargues......Page __sk_0248.djvu
11-3 The work of Pascal......Page __sk_0253.djvu
11-4 The principle of duality......Page __sk_0256.djvu
11-5 The relationship between projective and Euclidean geometries......Page __sk_0261.djvu
12-1 Descartes and Fermat......Page __sk_0264.djvu
12-2 The need for new methods in geometry......Page __sk_0267.djvu
12-3 The concepts of equation and curve......Page __sk_0270.djvu
12-4 The parabola......Page __sk_0278.djvu
12-5 Finding a curve from its equation......Page __sk_0283.djvu
12-6 The ellipse......Page __sk_0285.djvu
12-7 The equations of surfaces......Page __sk_0287.djvu
12-8 Four-dimensional geometry......Page __sk_0289.djvu
12-9 Summary......Page __sk_0291.djvu
13-1 Mastery of nature......Page __sk_0294.djvu
13-2 The search for scientific method......Page __sk_0295.djvu
13-3 The scientific method of Galileo......Page __sk_0298.djvu
13-4 Functions and formulas......Page __sk_0304.djvu
13-5 The formulas describing the motion of dropped objects......Page __sk_0307.djvu
13-6 The formulas describing the motion of objects thrown downward......Page __sk_0313.djvu
13-7 Formulas for the motion of bodies projected upward......Page __sk_0314.djvu
14-1 Introduction......Page __sk_0321.djvu
14-2 The concept of parametric equations......Page __sk_0322.djvu
14-3 The motion of a projectile dropped from an airplane......Page __sk_0324.djvu
14-4 The motion of projectiles launched by cannons......Page __sk_0327.djvu
14-5 The motion of projectiles fired at an arbitrary angle......Page __sk_0332.djvu
14-6 Summary......Page __sk_0337.djvu
15-1 The revolution in astronomy......Page __sk_0340.djvu
15-2 The objections to a heliocentric theory......Page __sk_0344.djvu
15-3 The arguments for the heliocentric theory......Page __sk_0345.djvu
15-4 The problem of relating earthly and heavenly motions......Page __sk_0348.djvu
15-5 A sketch of Newton\'s life......Page __sk_0350.djvu
15-6 Newton\'s key idea......Page __sk_0351.djvu
15-7 Mass and weight......Page __sk_0354.djvu
15-8 The law of gravitation......Page __sk_0355.djvu
15-9 Further discussion of mass and weight......Page __sk_0357.djvu
15-10 Some deductions from the law of gravitation......Page __sk_0360.djvu
15-11 The rotation of the earth......Page __sk_0366.djvu
15-12 Gravitation and the Keplerian laws......Page __sk_0369.djvu
15-13 Implications of the theory of gravitation......Page __sk_0373.djvu
16-2 The problems leading to the calculus......Page __sk_0379.djvu
16-3 The concept of instantaneous rate of change......Page __sk_0381.djvu
16-4 The concept of instantaneous speed......Page __sk_0382.djvu
16-5 The method of increments......Page __sk_0385.djvu
16-6 The method of increments applied to general functions......Page __sk_0388.djvu
16-7 The geometrical meaning of the derivative......Page __sk_0393.djvu
16-8 The maximum and minimum values of functions......Page __sk_0396.djvu
17-1 Differential and integral calculus compared......Page __sk_0402.djvu
17-2 Finding the formula from the given rate of change......Page __sk_0403.djvu
17-3 Applications to problems of motion......Page __sk_0404.djvu
17-4 Areas obtained by integration......Page __sk_0408.djvu
17-5 The calculation of work......Page __sk_0411.djvu
17-6 The calculation of escape velocity......Page __sk_0415.djvu
17-7 The integral as the limit of a sum......Page __sk_0418.djvu
17-8 Some relevant history of the limit concept......Page __sk_0423.djvu
17-9 The Age of Reason......Page __sk_0426.djvu
18-1 Introduction......Page __sk_0430.djvu
18-2 The motion of a bob on a spring......Page __sk_0431.djvu
18-3 The sinusoidal functions......Page __sk_0432.djvu
18-4 Acceleration in sinusoidal motion......Page __sk_0441.djvu
18-5 The mathematical analysis of the motion of the bob......Page __sk_0443.djvu
18-6 Summary......Page __sk_0448.djvu
19-1 Introduction......Page __sk_0450.djvu
19-2 The nature of simple sounds......Page __sk_0452.djvu
19-3 The method of addition of ordinates......Page __sk_0456.djvu
19-4 The analysis of complex sounds......Page __sk_0459.djvu
19-5 Subjective properties of musical sounds......Page __sk_0462.djvu
20-2 The historical background......Page __sk_0466.djvu
20-3 The mathematical content of Gauss\'s non-Euclidean geometry......Page __sk_0472.djvu
20-4 Riemann\'s non-Euclidean geometry......Page __sk_0474.djvu
20-5 The applicability of non-Euclidean geometry......Page __sk_0476.djvu
20-6 The applicability of non-Euclidean geometry under a new interpretation of line......Page __sk_0478.djvu
20-7 Non-Euclidean geometry and the nature of mathematics......Page __sk_0485.djvu
20-8 The implications of non-Euclidean geometry for other branches of our culture......Page __sk_0488.djvu
21-2 The applicability of the real number system......Page __sk_0492.djvu
21-3 Baseball arithmetic......Page __sk_0495.djvu
21-4 Modular arithmetics and their algebras......Page __sk_0498.djvu
21-5 The algebra of sets......Page __sk_0505.djvu
21-6 Mathematics and models......Page __sk_0511.djvu
22-1 Introduction......Page __sk_0513.djvu
22-2 A brief historical review......Page __sk_0514.djvu
22-3 Averages......Page __sk_0516.djvu
22-4 Dispersion......Page __sk_0517.djvu
22-5 The graph and the normal curve......Page __sk_0519.djvu
22-6 Fitting a formula to data......Page __sk_0525.djvu
22-7 Correlation......Page __sk_0530.djvu
22-8 Cautions concerning the uses of statistics......Page __sk_0532.djvu
23-1 Introduction......Page __sk_0536.djvu
23-2 Probability for equally likely outcomes......Page __sk_0538.djvu
23-3 Probability as relative frequency......Page __sk_0543.djvu
23-4 Probability in continuous variation......Page __sk_0544.djvu
23-5 Binomial distributions......Page __sk_0547.djvu
23-6 The problems of sampling......Page __sk_0552.djvu
24-2 The structure of mathematics......Page __sk_0555.djvu
24-3 The values of mathematics for the study of nature......Page __sk_0560.djvu
24-4 The aesthetic and intellectual values......Page __sk_0564.djvu
24-5 Mathematics and rationalism......Page __sk_0566.djvu
24-6 The limitations of mathematics......Page __sk_0567.djvu
Table of Trigonometric Ratios......Page __sk_0571.djvu
Answers to Selected and Review Exercises......Page __sk_0573.djvu
Additional Answers and Solutions......Page __sk_0583.djvu
Index......Page __sk_0647.djvu