Basic Calculus

Part I: From Archimedes to Newton

Introduction
The links Solutions, Corrections and Typos are pdf files that can be accessed with Acrobat Reader

1. The Greeks Measure the Universe
1. The Pythagoreans Measure Length 2. The Measure of Angles 3. Eratosthenes Measures the Earth 4. Right Triangles 5. Aristarchus Sizes up the Universe 6. The Sandreckoner 7. Postscript Exercises and Solutions Corrections and Typos	Pythagoras Eratosthenes Aristarchus Archimedes Stellar Parallax

2. Ptolemy and the Dynamics of the Universe
1. A Geometry of the Shadows of the Motion of the Sun 2. Geometry in the Almagest 3. The Solar Model 4. The Modern Perspective 5. Another Look at the Solar Model 6. Epicycles 7. Postscript Exercises and Solutions Corrections and Typos	Hipparchus Ptolemy Retrograde Motion with Epicycles

3. Archimedes Measures Area
1. The Conic Sections 2. The Question of Area 3. Playing with Squares 4. The Area of the Parabolic Section 5. The Method 6. Postscript Exercises and Solutions Corrections and Typos	Greek Texts in Translation Apollonius Archimedes More about Archimedes The Story of Archimedes's Manuscript

4. A New Astronomy and a New Geometry
1. A New Astronomy 2. The Studies of Galileo 3. The Geometry of Descartes 4. Circles and Trigonometry 5. The Ellipse 6. Cavalieri's Principle 7. Kepler's Analysis of the Orbits 8. The Method of Successive Approximations 9. Computing Orbital Information 10. Postscript Exercises and Solutions Corrections and Typos	Copernicus Brahe Kepler Galileo Descartes Cavalieri The Planets Planetary Data Kepler's Equation and the Rudolphine tables Computer Model of Elliptical Orbits Generated by Kepler's Equations

5. The Calculus of Leibniz
1. Straight Lines 2. Tangent Lines to Curves 3. Areas and Differentials 4. The Fundamental Theorem of Calculus 5. Functions a. The Derivative b. Antiderivatives 6. Some Applications a. Finding Maximum and Minimum Values b. Volumes c. Lengths of Curves 7. Postscript Exercises and Solutions Corrections and Typos	Leibniz Additional Sources for Leibniz Leibniz's Role in the Developement of Calculus

6. The Calculus of Newton
1. Areas under Simple Curves 2. The Fundamental Theorem (again) 3. Computing Definite Integrals 4. Moving Points 5. The Trajectory of a Projectile 6. Application to Ballistics? 7. Postscript Exercises and Solutions Corrections and Typos	Newton Newton's Role in the Development of Calculus Title Page of "Analysis per Quantitatum ..."

7. The Principia
1. Equal Areas in Equal Times 2. Analyzing Centripetal Force 3. The Inverse Square Law 4. Test Case: The Orbit of the Moon 5. The Law of Universal Gravitation 6. Incredible Consequences 7. Postscript Exercises and Solutions Corrections and Typos	Title Page of the Principia Newton's Work on Orbits and Gravitation The "Newton" One Pound Note Facts about the Solar System Planetary Data Galileo Probe Space Elevator