§ 1. Foundation of the doctrine of chances, as taught by mathematics 61 2. The doctrine tenable 63 3. On what foundation it really rests 64 4. Its ultimate dependence on causation 68 5. Theorem of the doctrine of chances which relates to the cause of a given event 72 6. How applicable to the elimination of chance 74 CHAPTER XIX. _Of the Extension of Derivative Laws to Adjacent Cases._ § 1. Derivative laws, when not casual, are almost always contingent on collocations 78 2. On what grounds they can be extended to cases beyond the bounds of actual experience 80 3. Those cases must be adjacent cases 82 CHAPTER XX. _Of Analogy._ § 1. Various senses of the word analogy 86 2. Nature of analogical evidence 87 3. On what circumstances its value depends 91 CHAPTER XXI. _Of the Evidence of the Law of Universal Causation._ § 1. The law of causality does not rest on an instinct 95 2. But on an induction by simple enumeration 100 3. In what cases such induction is allowable 102 4. The universal prevalence of the law of causality, on what grounds admissible 105 CHAPTER XXII. _Of Uniformities of Coexistence not dependent on Causation._ § 1. Uniformities of coexistence which result from laws of sequence 110 2. The properties of Kinds are uniformities of coexistence 111 3. Some are derivative, others ultimate 113 4. No universal axiom of coexistence 114 5. The evidence of uniformities of coexistence, how measured 117 6. When derivative, their evidence is that of empirical laws 117 7. So also when ultimate 119 8. The evidence stronger in proportion as the law is more general 120 9. Every distinct Kind must be examined 121 CHAPTER XXIII. _Of Approximate Generalizations, and Probable Evidence._ § 1. The inferences called probable, rest on approximate generalizations 124 2. Approximate generalizations less useful in science than in life 124 3. In what cases they may be resorted to 126