A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia

1. Introduction.- 1.1 A Brief History of th e Infinitesimal.- 1.2 The Principia and its Methods.- 1.3 On Nonstandard Analysis.- 1.4 Objectives.- 1.5 Achieving our Goals.- 1.6 Organisation of this Book.- 2. Geometry Theorem Proving.- 2.1 Historical Background.- 2.2 Algebraic Techniques.- 2.3 Coordinate-Free Techniques.- 2.4 Formalizing Geometry in Isabelle.- 2.5 Concluding Remarks.- 3. Constructing the Hyperreals.- 3.1 Isabelle/HOL.- 3.2 Propertiesof an Infinitesimal Calculus.- 3.3 Internal Set Theory.- 3.4 Constructions Leading to the Reals.- 3.5 Filters and Ultrafilters.- 3.6 Ultrapower Construction of the Hyperreals.- 3.7 Structure of the Hyperreal Number Line.- 3.8 The Hypernatural Numbers.- 3.9 An Alternative Construction for the Reals.- 3.10 Related Work.- 3.11 Concluding Remarks.- 4. Infinitesimal and Analytic Geometry.- 4.1 Non-Archimedean Geometry.- 4.2 New Definitions and Relations.- 4.3 Infinitesimal Geometry Proofs.- 4.4 Verifying the Axioms of Geometry.- 4.5 Concluding Remarks.- 5. Mechanizing Newton’s Principia.- 5.1 Formalizing Newton’s Properties.- 5.2 Mechanized Propositions and Lemmas.- 5.3 Ratios of Infinitesimals.- 5.4 Case Study : Propositio Kepleriana.- 6. Nonstandard Real Analysis.- 6.1 Extending a Relation to the Hyperreals.- 6.2 Towards an Intuitive Calculus.- 6.3 Real Sequences and Series.- 6.4 Some Elementary Topology of the Reals.- 6.5 Limits and Continuity.- 6.6 Differentiation.- 6.7 On the Transfer Principle.- 6.8 Related Work and Conclusions.- 7. Conclusions.- 7.1 Geometry, Newton , and the Principia.- 7.2 Hyperreal Analysis.- 7.3 Further Work.- 7.4 Concluding Remarks.