The Key to Newton's Dynamics
  PREFACE
  ACKNOWLEDGMENTS

 collapse sectionPART ONE—  THE BACKGROUND TO NEWTON'S SOLUTION
 collapse sectionOne—  A Simplified Solution:  The Area Law, the Linear Dynamics Ratio, and the Law of Gravitation
 collapse sectionA Simplified Solution
 •Theorem 1
 •Theorem 3
 •Problem 3
 •The Reception
 collapse sectionTwo—  An Overview of Newton's Dynamics:  The Problem of the Planets and the Principia
 •The Direct Problem
 •Newton's Debt to Descartes
 •Kepler's Law of Equal Areas and the Polygonal Approximation
 •Newton's Debt to Galileo
 •The Parabolic Approximation and the Linear Dynamics Ratio
 •Elliptical Orbits and the Parabolic Approximation
 •The Circular Approximation and the Circular Dynamics Ratio
 •Conclusion
 collapse sectionThree—  Newton's Early Dynamics:  On Uniform Circular Motion
 •Uniform Circular Motion:  The Polygonal Approximation
 collapse sectionUniform Circular Motion:  The Waste Book
 •Square
 •Polygon
 •Circle
 •An Alternate Demonstration
 •Uniform Circular Motion:  The Parabolic Approximation
 •On Circular Motion
 •Elliptical Motion:  The Circular Approximation
 •Elliptical Motion
 •Conclusion

 collapse sectionPART TWO—  A GUIDED STUDY TO NEWTON'S SOLUTION
 collapse sectionFour—  The Paradigm Constructed: On Motion , Theorems 1, 2, and 3
 collapse sectionIntroduction:  Definitions, Hypotheses, and Lemmas
 •Definitions
 •Hypotheses
 •Lemmas
 •Hypothesis 3 Becomes Lemma 1
 collapse sectionTheorem 1—  The Law of Equal Areas in Equal Times
 •Demonstration
 collapse sectionTheorem 2—  Uniform Circular Motion
 •Demonstration
 collapse sectionTheorem 3—  The Linear Dynamics Ratio
 •Demonstration
 •Conclusion
 collapse sectionFive—  The Paradigm Applied: On Motion , Problems 1, 2, and 3
 collapse sectionProblem 1—  A Circular Orbit with the Center of Force on the Circumference of the Circle
 •Step 1—  The Diagram
 •Step 2—  The Analysis
 •Step 3—  The Limit
 •Conclusion
 collapse sectionProblem 2—  An Elliptical Orbit with the Center of Force at the Center of the Ellipse
 •Step 1—  The Diagram
 •Step 2—  The Analysis
 •Step 3—  The Limit
 •Conclusion
 collapse sectionProblem 3—  An Elliptical Orbit with the Center of Force at a Focus of the Ellipse
 •Step 1—  The Diagram
 •Step 2—  The Analysis ([2-B] to [2-Q])
 •Step 3—  The Limit
 •Conclusion
 •Conclusion
 collapse sectionSix—  The Paradigm Extended: On Motion , Theorem 4 and Problem 4
 •Theorem 4—  The Three-Halves Power Law
 •Problem 4—  Given an Inverse Square Force, Find the Resulting Conic Section
 •Book Three  Proposition 13  Theorem 13
 •Conclusion

 collapse sectionPART THREE—  THE REVISIONS AND EXTENSIONS TO NEWTON'S SOLUTION
 collapse sectionSeven—  The Principia and its Relationship to On Motion :  A Reference Guide for the Reader
 •Introduction
 •Definitions
 •Axioms or Laws of Motion
 •Section 1
 •Lemma 11—  A Detailed Discussion
 •Section 2
 collapse sectionProposition 9—  A Detailed Discussion
 •Demonstration
 •Lemma 12 and Proposition 10
 •Section 3
 •Conclusion
 collapse sectionEight—  Newton's Unpublished Proposed Revisions:  Two New Methods Revealed
 •Proposed Propositions 6, 7, and 8
 •The Three Techniques
 •The Comparison Theorem
 •The Proposed Propositions 9 and 10
 •The Locke Solution
 •Conclusion
 collapse sectionNine—  Newton's Published Recast Revisions:  Two New Methods Concealed
 •The Revision of Lemma 11
 •The Revision of Proposition 1
 •The Revision of Proposition 6
 •Proposition 6—  Theorem 5
 •A Rejected Revision of Proposition 6
 •The Revision of Proposition 7—  An Alternate Solution
 •The Revision of Proposition 9—  An Alternate Solution
 •The Revision of Proposition 10—  An Alternate Solution
 collapse sectionThe Revision of Proposition 11—  Alternate Solutions
 •An Unpublished Alternate Solution
 •The Published Alternate Solution
 •Conclusion
 collapse sectionTen—  Newton's Dynamics in Modern Mathematical Dress:  The Orbital Equation and the Dynamics Ratios
 •The Orbital Equation and Curvature
 •The Dynamics Ratios and the Orbital Equation
 •The Revision of Corollary 1 of Proposition 13
 •A Detailed Solution of the Inverse Problem
 •Conclusion

 collapse sectionAPPENDIX
 •An English Translation of Sections 1, 2, and 3 of Book One from the First (1687) Edition of Newton's Mathematical Principles of Natural Philosophy
 •Preface to the Reader
 •On this Work of Mathematical Physics  Of the Most Outstanding Man Most Learned Isaac Newton
 collapse sectionOn the Motion of Bodies Book One
 collapse sectionSection 1
 •Lemma 1
 •Lemma 2
 •Lemma 3
 •Lemma 4
 •Lemma 5
 •Lemma 6
 •Lemma 7
 •Lemma 8
 •Lemma 9
 •Lemma 10
 •Lemma 11
 •Scholium
 collapse sectionSection 2
 •Proposition 1—  Theorem 1
 •Proposition 2—  Theorem 2
 •Scholium
 •Proposition 3—  Theorem 3
 •Scholium
 •Proposition 4—  Theorem 4
 •Scholium
 •Proposition 5—  Problem 1
 •Proposition 6—  Theorem 5
 •Proposition 7—  Problem 2
 •Proposition 8—  Problem 3
 •Scholium
 •Proposition 9—  Problem 4
 •Lemma 12
 •Proposition 10—  Problem 5
 •Scholium
 collapse sectionSection 3
 •Proposition 11—  Problem 6
 •Proposition 12—  Problem 7
 •Lemma 13
 •Lemma 14
 •Proposition 13—  Problem 8
 •Proposition 14—  Theorem 6
 •Proposition 15—  Theorem 7
 •Proposition 16—  Theorem 8
 •Proposition 17—  Problem 9
 collapse sectionNotes
 •One— A Simplified Solution: The Area Law, the Linear Dynamics Ratio, and the Law of Gravitation
 •Two— An Overview of Newton's Dynamics: The Problem of the Planets and the Principia
 •Three— Newton's Early Dynamics: On Uniform Circular Motion
 •Four— The Paradigm Constructed:On Motion , Theorems 1, 2, and 3
 •Five— The Paradigm Applied:On Motion , Problems 1, 2, and 3
 •Six— The Paradigm Extended:On Motion , Theorem 4 and Problem 4
 •Seven— The Principia and its Relationship to On Motion : A Reference Guide for the Reader
 •Eight— Newton's Unpublished Proposed Revisions: Two New Methods Revealed
 •Nine— Newton's Published Recast Revisions: Two New Methods Concealed
 •Ten— Newton's Dynamics in Modern Mathematical Dress: The Orbital Equation and the Dynamics Ratios
 •An English Translation of Sections 1, 2, and 3 of Book One from the First (1687) Edition of Newton's Mathematical Principles of Natural Philosophy
 collapse sectionREFERENCES
 •Primary Works
 •Secondary Works
 collapse sectionINDEX TO THE GUIDED STUDY AND THE TRANSLATION
 •Uniform Circular Motion
 •On Motion (On Motion of Bodies in Orbit)
 collapse sectionPrincipia (The Mathematical Principles of Natural Philosophy)
 •Book One
 •Book Three
 •Proposed Radical Revisions
 collapse sectionGENERAL INDEX
 •A
 •B
 •C
 •D
 •E
 •F
 •G
 •H
 •I
 •J
 •K
 •L
 •M
 •N
 •O
 •P
 •R
 •S
 •T
 •U
 •V
 •W
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