the other Extreme (the Major) belongs to the Middle.” By the Middle Term, in the Aristotelian phraseology, is to be understood a certain Class of things; by the Major Extreme an Attribute ascribed, or to be ascribed, to that class; and by the Minor Extreme a Portion or Portions of the same class. Induction, therefore, as explained by Aristotle, is this: the examining the several portions of a given class of things; the finding that they each possess a certain attribute other than the attribute which constitutes the class; and the inferring that the observed attribute belongs to the members of that class universally. The following, for example, would be an Aristotelian induction: The capability of combining with oxygen is a property of iron, of copper, of gold, of silver, of lead, of tin, etc.; and these constitute the whole class Metal; therefore, whatever is a metal has the property of combining with oxygen. In another portion of Aristotle's works the following explanation is given (Top. I. 12): “Induction is the proceeding from the singulars to the universal. E.g., let us suppose that the person who has knowledge is best as a pilot, and as a charioteer; and, universally, in each department the person who has knowledge is the most excellent.” ^*

The only induction adequately analysed by the ancients was that in which all the portions of the class, respecting which the inference is to be made, are supposed to be known by direct experience. But in actual practice the data of induction are of course very rarely capable of being thus exhaustive; and indeed the word Induction is now specially appropriated to what the old logicians called Incomplete Induction, viz, that in which only a portion of the class in question is experientially known, which portion is assumed to be a sufficient specimen of the whole.

The title, Father of Inductive Philosophy, has been frequently awarded to Francis Bacon. The designation is not quite appropriate, nor yet altogether unfitting. Bacon did not himself extend the domain of experimental science; he did not appreciate duly what some investigators of now illustrious renown had a short time previously effected; his exposition of the methods of investigation was unavoidably indefinite, and it is now perceived to be, in certain most important respects, essentially erroneous. What Bacon did was this: he descried, he deeply felt, and with majestic eloquence he proclaimed and enforced the principle that the great want of mankind in his age, as to science and art, was the extensive observation and strict questioning of nature; with

a prophetic exaltation of spirit he stood singly on a Mount Pisgah of science, and vividly enjoyed and depicted a glorious vision of rich achievements to be accomplished; he taught that the way to realize such grand results was, in the first place, to tread a humble path of patient inquiry; and, lastly, he attempted, though he could not succeed in the construction a priori of a logic of experimental discovery.

Without injustice to either Aristotle or Bacon we may say that the Logic of Scientific Investigation is a recent addition to mental science. It has been hitherto developed exclusively by English thinkers. Archbishop Whately was the first who indicated a genuine analysis of the inductive generalization. Dr. Whewell is the historian of the Inductive Sciences; and he has copiously treated of their philosophy. He did, doubtless, exaggerate the prowess of the Intellect in its relation to Experience; but it may be fairly questioned whether his critics of the opposite school have not erred as widely in the contrary direction. It is characteristic of the tendency of Dr. Whewell's mind, that he suggested the avowed innovation of giving a new definition to the word Induction, and assigning a new reference to its etymology. He urges that the inductive result is obtained by means of a mental conception being superinduced upon the thereby colligated facts—as in Kepler's discovery of the orbit of the planet Mars being the periphery of an ellipse; and by Induction Dr. Whewell would have us to understand this superinducing of the appropriate conception. ^* The suggestion, I need scarcely say, has not been adopted. We must not omit mentioning here, though it will suffice to merely mention, Sir John Herschel's instructive “Discourse on the Study of Natural Philosophy.” John Stuart Mill, who has recently passed away while still fully exercising the matured vigor of his profound intellect, accomplished the task which Bacon somewhat prematurely attempted, and which has been regarded by many as impracticable—that of constructing a logic of scientific investigation. Mr. Mill defines Induction thus (System of Logic, III. ii. 1): That operation of the mind by which we infer, that what we know to be true in a particular case or cases, will be true in all cases, which resemble the former in certain assignable respects. In other words,” it is added, “Induction is the process by which we conclude, that what is true of certain individuals of a class is true of the whole class, or that what is true at certain times, will be true under similar circumstances at all times.” In another passage the same author says (III. iii. 1): “Induction properly so called… may, then, be summarily defined as Generalization from Experience. It consists in inferring from some individual instances, in which a phenomenon is observed to occur, that it occurs in all instances of a certain class; namely,

in all which resemble the former, in what are regarded as the material circumstances.”

In this definition of Induction Mr. Mill is in unison with Aristotle and other ancient writers, and with the popular reviver of Logic, Archbishop Whately. The definitions given by more recent expositors are tantamount. Induction being thus defined, let us now notice explicitly, in order to decisive clearness, some of the properties of Induction which are expressed or implied in that definition.

• transcend the region of the Probable—the term probable being here employed, not in the more restricted sense of the popular usage, but in the wider acceptation whereby it includes moral certainty. That all matter throughout the universe gravitates; that when the Sun shall rise to-morrow, his disc will be circular; that of the liquid in an unknown river, the main ingredients are oxygen and hydrogen; that no human being now alive will survive till the year 2073: all these are propositions as to which it may be presumed that no person would entertain any doubt; but yet we may distinguish degrees of difference in our assurance of them. They are results of accumulated evidence; and “probable evidence,” as Bishop Butler justly observes, is that which “admits of degrees, and of all variety of them from the highest moral certainty to the very lowest presumption.”

It is, then, an essential characteristic of inductive inference, that it does not lead to absolute knowledge, though it can. attain to what is practically equivalent—moral certainty. This essential characterestic, however, has been, strange to say, ignored in the discussion of some of the fundamental questions pertaining to the philosophy of Induction; and yet, by due attention to it, we may obtain, I am persuaded, the solution of that peculiar and various amount of unsatisfactory paradox which still disfigures the basis of this department of sciences.

Inductive inference, as we have seen, conducts to general truths. We now, therefore, proceed to ask, are all general truths inductive? In other words, are there any universal propositions which we know to be true, and which are such that our knowledge, whether of those propositions themselves, or of those from which they are deduced, is not dependent upon the fact of a verifying test having been applied in a sufficiency of individual instances?

Let us take the arithmetical proposition, the square of 3 plus the square of 4 is the square of 5. This proposition is unquestionably true. And it is a general or universal proposition; for the assertion virtually is, that in any instance whatever of there being an aggregate of things the number of which is the square of 3, and also an aggregate of things the number of which is the square of 4, then in the sum of the two aggregates the number of those things is the square of 5. Now, what is the foundation of our certainty that this proposition is true? It may well be that none of us has ever formally tested the truth of the proposition by actual experiment; and yet none of us doubts it. It is deduced from other propositions; and we are now to see whether those others are inductive. The expression, the square of 3, means 3 taken 3 times; and so also as to 4 and 5. But what do we mean by these words, three, four, five, etc.? As the word two simply means 1 + 1, so the word three means 2 + 1; and so on until we have defined ten. And, further, the word eleven means 10 + 1; the word twelve means 10 + 2; and so on until we have defined

twenty. These definitions being assumed, it is demonstrable by the mere substitution of equivalent names, and so performing the enumeration in the abstract, that the square of 3 is 9, that the square of 4 is 16, that 9 + 16 is 25, and that the square of 5 is the same. But those fundamental assumptions, viz., that two means 1 + 1, that three means 2 + 1, etc., are not inductive propositions; they are not merely results ascertained from repeated experiment. Our knowledge of the proposition that five means 4 + 1 is not a moral certainty accruing from accumulated evidence. These propositions are definitions, and nothing more. We find, therefore, that the proposition, the square of 3 + the square of 4 is the square of 5, is a universal proposition, the truth of which we know; and that our knowledge of its truth is not dependent upon Induction. And a similar analysis might be applied to all propositions whatever belonging to the pure science of Number.

But it may be said: Definitions are simply explanations of words; and how, then, can deductions from mere definitions demonstrate an objective matter of fact? We must answer that they cannot do so; such a result is impossible. That there are in real existence aggregates of things severally numbering 9, and also aggregates of things severally numbering 16, as to such propositions we have no absolute knowledge further than may be afforded to us by direct perception of individual things. But we have an absolute knowledge of this, that if there exist any where an aggregate of things whose number is 9, and also an aggregate of things whose number is 16, in that case the sum of those things numbers 25. Such a proposition comprises a condition as to the supposition of real existence; but we know absolutely the truth of the conditional proposition, which truth consists in this, that the predicate is coextensive with the subject. It may be, or may not be, a matter of fact that there is at the present moment in St. Giles's, London, a mendicant having two pockets, in one of which are four, and in the other three, penny pieces. But it is a proposition of absolute certainty, that if there be such a personage so circumstanced he has in his pockets coins to the number of seven. All general propositions asserting unconditionally any matter of fact, or any real existence, are, or ought to be, derived from inductive data: but there are also general propositions expressly or virtually hypothetic; and many of these, in various departments of knowledge, are necessary truths, that is, propositions whose truth is known absolutely, and not merely probable or certain according to the amount of experience.

We are now in a position enabling us to obtain a more distinct view of what we mean by a necessary truth. A necessary truth is, of course, a species of proposition or assertion: and all propositions, of whatsoever kind they may be, and however much they may vary in other respects, agree in this, that every proposition consists of a subject and a predicate; that is, a subject-

thing is spoken of, and to that subject-thing there is ascribed an attribute. What, then, are we to understand by a proposition being self-evident? It must be this, that the perception of the subject-thing comprises the perception of the predicated attribute; that we cannot perceive the one without perceiving the other. In other words, a self-evident proposition is such that no adequate definition or explanation of the subject of the proposition can be given without its comprising the ascription of the predicated attribute. If, on the other hand, we take any inductive conclusion—e.g., All matter gravitates, All animals having horns on the forehead are ruminant, All revolutionary democracies tend to military monarchy—in such a case we find that the predicated attribute is additional to, outside of, the definition of the subject. But if we take the proposition, 4 is 3 + 1, the mere definition of what we mean by four gives us the predicate. The attribute, in such an instance, is not additional to, but is included in, the definition of the subject. Again, let us take the axiom, If equals be added to equals, the sums are equal: we cannot give any definition or explanation of what is meant by equals, and of what is meant by adding, without implying the equality of the sums. Let us, for further illustration, refer to the science of quantity in space. The fundamental axioms peculiar to Geometry are two, of which the first is, that Two straight lines cannot enclose space. This proposition may be resolved into two alternative propositions. Any two lines must either meet one another, or not. As to the first alternative, it is obvious that the perception of two lines not meeting one another comprises the perception of their not enclosing space. And so also as to the second alternative: we cannot have the perception of two lines diverging from a point, and of each of those lines being straight, without our perceiving that the divergence is perpetual. The term “straight line” may perhaps be considered to be undefinable as being the expression of a simple conception; but we may doubtless explain the conception, if requisite, by variation of language, or by whatever means of illustration. We cannot, however, give any explanation of straightness, nor any definition or explanation of “diverging,” without implying the perpetuity of the divergence.

In like manner we might explain why the following propositions are selfevident: Two diverging straight lines are not parallel to the same third; Every event has a cause; Causes perfectly similar produce perfectly similar effects.

Such, then, is our definition of self-evident truth, viz, A self-evident proposition is a proposition such that the perception of the subject-thing comprises the perception of the predicated attribute; or, it is a proposition such that any adequate definition or explanation of the subject implies the predicate. Having this definition, we have no difficulty in completing our definition of Necessary Truth. A necessary truth is that which is either a self-evident