4–6 Applications of the number system2–2 Mathematics in early civilizations
2–3 The classical Greek period
2–4 The Alexandrian Greek period
2–8 Developments from 1550 to 1800
2–9 Developments from 1800 to the present
2–10 The human aspect of mathematics
3–2 The concepts of mathematics
3–7 The creation of mathematics
4 Number: the Fundamental Concept
4–2 Whole numbers and fractions
4–5 The axioms concerning numbers
4–6 Applications of the number system
5 Algebra, the Higher Arithmetic
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
6–7 The cultural influence of Euclidean geometry
7 Charting the Earth and the Heavens
7–2 Basic concepts of trigonometry
7–3 Some mundane uses of trigonometric ratios
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
9–1 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
10–2 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
11–1 The problem suggested by projection and section
11–5 The relationship between projective and Euclidean geometries
12–2 The need for new methods in geometry
12–3 The concepts of equation and curve
12–5 Finding a curve from its equation
12–7 The equations of surfaces
12–8 Four-dimensional geometry
13 The Simplest Formulas in Action
13–2 The search for scientific method
13–3 The scientific method of Galileo
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 Curvilinear Motion
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
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
15–9 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
16–2 The problems leading to the calculus
16–3 The concept of instantaneous rate of change
16–4 The concept of instantaneous speed
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
17–1 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
17–6 The calculation of escape velocity
17–7 The integral as the limit of a sum
17–8 Some relevant history of the limit concept
18 Trigonometric Functions and Oscillatory Motion
18–2 The motion of a bob on a spring
18–4 Acceleration in sinusoidal motion
18–5 The mathematical analysis of the motion of the bob
19 The Trigonometric Analysis of Musical Sounds
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
20–2 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
21–2 The applicability of the real number system
21–4 Modular arithmetics and their algebras
22 The Statistical Approach to the Social and Biological Sciences
22–2 A brief historical review
22–5 The graph and the normal curve
22–6 Fitting a formula to data
22–8 Cautions concerning the uses of statistics
23–2 Probability for equally likely outcomes
23–3 Probability as relative frequency
23–4 Probability in continuous variation
24 The Nature and Values of Mathematics
24–2 The structure of mathematics
24–3 The values of mathematics for the study of nature
24–4 The aesthetic and intellectual values
24–5 Mathematics and rationalism
24–6 The limitations of mathematics
Answers to Selected and Review Exercises