§ 1. For what purpose trains of reasoning exist 234 2. A train of reasoning is a series of inductive inferences 234 3. --from particulars to particulars through marks of marks 237 4. Why there are deductive sciences 240 5. Why other sciences still remain experimental 244 6. Experimental sciences may become deductive by the progress of experiment 246 7. In what manner this usually takes place 247 CHAPTER V. _Of Demonstration, and Necessary Truths._ § 1. The Theorems of geometry are necessary truths only in the sense of necessarily following from hypotheses 251 2. Those hypotheses are real facts with some of their circumstances exaggerated or omitted 255 3. Some of the first principles of geometry are axioms, and these are not hypothetical 256 4. --but are experimental truths 258 5. An objection answered 261 6. Dr. Whewell's opinions on axioms examined 264 CHAPTER VI. _The same Subject continued._ § 1. All deductive sciences are inductive 281 2. The propositions of the science of number are not verbal, but generalizations from experience 284 3. In what sense hypothetical 289 4. The characteristic property of demonstrative science is to be hypothetical 290 5. Definition of demonstrative evidence 292 CHAPTER VII. _Examination of some Opinions opposed to the preceding doctrines._ § 1. Doctrine of the Universal Postulate 294 2. The test of inconceivability does not represent the aggregate of past experience 296 3. --nor is implied in every process of thought 299 4. Sir W. Hamilton's opinion on the Principles of Contradiction and Excluded Middle 306 BOOK III. OF INDUCTION. CHAPTER I. _Preliminary Observations on Induction in general._ § 1. Importance of an Inductive Logic 313 2. The logic of science is also that of business and life 314 CHAPTER II. _Of Inductions improperly so called._ § 1. Inductions distinguished from verbal transformations 319 2. --from inductions, falsely so called, in mathematics 321 3. --and from descriptions 323 4. Examination of Dr. Whewell's theory of Induction 326 5. Further illustration of the preceding remarks 336 CHAPTER III. _On the Ground of Induction._