Chapter IV. Of Trains of Reasoning, and Deductive Sciences. § 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 [Pg xv] 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 [Pg xvi] Chapter III. On the Ground of Induction. § 1. Axiom of the uniformity of the course of nature 341 2. Not true in every sense. Induction per enumerationem simplicem 346 3. The question of Inductive Logic stated 348 Chapter IV. Of Laws of Nature. § 1. The general regularity in nature is a tissue of partial regularities, called laws 351 2. Scientific induction must be grounded on previous spontaneous inductions 355 3. Are there any inductions fitted to be a test of all others? 357