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'Mathematics, taught and learned appropriately, improves the mind and implants good habits of thought.' This tenet underlies all of Professor Polya's works on teaching and problem-solving. This book captures some of Pуlya's excitement and vision. In it he provides enlightenment for all those who have ever wondered how the laws of nature were worked out mathematically. The distinctive feature of the present book is the stress on the history of certain elementary chapters of science; these can be a source of enjoyment and deeper understanding of mathematics even for beginners who have little, or perhaps no, knowledge of physics.
Introduction
From the History of Astronomy: Measurement and Successive Approximation
Measurement
The Tunnel
Measuring: Triangulating
How Far Away is the Moon?
Why Teach Triangulation?
Astronomical measurements
Aristarchus of Samos
Radius of Earth: Eratosthenes
Rival Cosmologies
The Orbit of Venus
Tycho Brahe and Kepler
The Mars year
The Orbit of Mars
A Word to the Perceptive Reader
Newton’s Problem of a Comet’s Path
Successive approximation
First Application
Extraction of SquareRoots
Newton’s method of successive approximation
The General Method of Newton
√a
∛a
5√a
From the History of Statics
Stevinus and Archimedes
Inclined Plane
Lever
Vectors
Inclined Plane
Pulley
Lever
Archimedes' Application of his Law of the Lever
( - ) * ( - ) = ( + )
Von Mises. Flight Triangle
From the History of Dynamics
Galileo
Heavier Bodies Fall Faster?
Not "Why?", But "How?"
How do Heavy Bodies Fall?
Dynamics of the Inclined Plane
Conservation of Energy
Law of Inertia
A Cannon Ball's Trajectory
Newton
Apples, Cannon Balls, and the Moon
Never Smoke Without Fire
That the Planets do Accelerate Towards the Sun
What is the Law of Universal Gravitation?
Uniform Circular Motion: Hamilton's Hodograph
On Newton's Discovery of the Law of Universal Gravitation
Scientific Attitude: Verification
Hindsight and Foresight
The Pendulum
The Dimensions Test
Simple Pendulum's Time of Swing
Determination of g by Pendulum Experiment
The Conical Pendulum
Escape velocity
Go-Around Velocity
Apropos Go-Away Velocity
The Force of Gravity
That Kepler's Third Law is a Newton's Law of Gravitation
Consequence
Planetary Mass
Go-Away Velocity
Ratio of Escape and Orbital Velocities
Physical Reasoning in Mathematics
Differential Equations and Their Use in Science
First examples
Rotating Fluid
Galileo: Free Fall
Catenary
Fall with Friction
Approximate formulae: Power series
Introduction
Calculation of ∛28
Fall with Friction Again
How Deep is a Well?
Pendulum: Small Oscillations
Physical analogy
What is a differential equation?
Example
Vector Fields
Direction Fields