John Stuart Mill on Inductive reasoning

“Induction... is a process of inference; it proceeds from the known to the unknown; and any operation involving no inference, any process in which what seems the conclusion is no wider than the premises from which it is drawn, does not fall within the meaning of the term. ...A general proposition is one in which the predicate is affirmed or denied of an unlimited number of individuals; namely, all, whether few or many, existing or capable of existing, which possess the properties connoted by the subject of the proposition.”

“A complete logic of the sciences would be also a complete logic of practical business and common life. Since there is no case of legitimate inference from experience, in which the conclusion may not legitimately be a general proposition; an analysis of the process by which general truths are arrived at, is virtually an analysis of all induction whatever. Whether we are inquiring into a scientific principle or into an individual fact, and whether we proceed by experiment or by ratiocination, every step in the train of inferences is essentially inductive, and the legitimacy of the induction depends in both cases upon the same conditions.”

“The first and second editions of the "System of Logic" contained a passage which purported to controvert the views of Archbishop Whately respecting Inductive Syllogism. In the third edition the most controversial portions of this passage are omitted, and additions are made, which materially modify the result. But it is still implied, that "Archbishop Whately's must be held to be the correct account" of no more than "the immediate major-premise in every inductive argument;" and it is still maintained, that "if we throw the whole course of any inductive argument into a series of syllogisms, we shall arrive, by more or fewer steps, at an ultimate syllogism, which will have for its major-premiss the principle, or axiom, of the uniformity of the course of nature;" which principle or axiom is regarded by Mr. Mill as known to us only by "induction."”

“Although... there is not yet extant a body of Inductive Logic, scientifically constructed; the materials for its construction exist, widely scattered, but abundant: and the selection and arrangement of those materials is a task with which intellects of the highest order, possessed of the necessary acquirements, have at length consented to occupy themselves. Within a few years three writers, profoundly versed in every branch of physical science, and not unaccustomed to carry their speculations into still higher regions of knowledge, have made attempts, of unequal but all of very great merit, towards the creation of a Philosophy of Induction: Sir John Herschel, in his [A Preliminary] Discourse on the Study of Natural Philosophy; Mr. Whewell, in his History and Philosophy of the Inductive Sciences; and, greatest of all, M. Auguste Comte, in his Cours de Philosophic Positive, a work which only requires to be better known, to place its author in the very highest class of European thinkers. That the present writer does not consider any of these philosophers, or even all of them together, to have entirely accomplished this important work, is implied in his attempting to contribute something further towards its achievement...”

“There are... in mathematics, some examples of so called induction, in which the conclusion does bear the appearance of a generalization grounded upon some of the particular cases included in it. A mathematician, when he has calculated a sufficient number of the terms of an algebraical or arithmetical series to have ascertained what is called the law of the series, does not hesitate to fill up any number of the succeeding terms without repeating the calculations. But I apprehend he only does so when it is apparent from à priori considerations (which might be exhibited in the form of demonstration) that the mode of formation of the subsequent terms, each from that which preceded it, must be similar to the formation of the terms which have been already calculated. And when the attempt has been hazarded without the sanction of such general considerations, there are instances upon record in which it has led to false results. ...Even, therefore, such cases as these, are but examples of what I have called induction by parity of reasoning, that is, not really induction, because not involving any inference of a general proposition from particular instances. ...I am happy to be able to refer, in confirmation of this view of what is called induction in mathematics, to the highest English authority on the philosophy of algebra, Mr. Peacock. See pp. 107-8 of his profound Treatise on Algebra.”