laws, we are necessitated to reason from an accumulation of particular instances to an universal rule."-Mansel, Prolegom. Log., p. 209.

Principle of Induction.-By the principle of induction is meant the ground or warrant on which we conclude that what has happened in certain cases, which have been observed, will also happen in other cases, which have not been observed. This principle is involved in the words of the wise man, Eccles. i. 9, "The thing that hath been, it is that which shall be: and that which is done is that which shall be done." In nature there is nothing insulated. All things exist in consequence of a sufficient reason, all events occur according to the efficacy of proper causes. In the language of Newton, Effectuum naturalium ejusdem generis eædem sunt causæ. The same causes produce the same effects. The principle of induction is an application of the principle of casuality. Phenomena have their proper causes, and these causes operate according to a fixed law. This law has been expressed by saying, substance is persistent. Our belief in the established order of nature is a primitive judgment, according to Dr. Reid and others, and the ground of all the knowledge we derive from experience. According to others this belief is a result or inference derived from experience.

4. From Whewell's Novum Organon Renovatum, 3d edition, 1858, p. 139:

The Pure Mathematical Sciences can hardly be called Inductive Sciences. Their principles

are not obtained by Induction from Facts, but are necessarily assumed in reasoning upon the subject-matter which those sciences involve.

5. From English Cyclopædia, edition 1867. Induction (αywyn), as defined by Archbishop Whately, is “a kind of argument which infers respecting a whole class what has been ascertained respecting one or more individuals of that class." According to Sir William Hamilton the word has been employed to designate three very different operations:-1. The objective process of investigating particular facts, as preparatory to Induction, which he observes is manifestly not a process of reasoning of any kind; 2. A material illation of a universal from a singular, as warranted either by the general analogy of nature or the special presumptions afforded by the object matter of any real science; 3. A formal illation of a universal from the individual, as legitimated solely by the laws of thought and abstracted from the conditions of any particular matter. The second of these operations is the inductive method of Bacon, which proceeds by means of rejections and conclusions, so as to arrive at those axioms or general laws from which we may infer by way of synthesis other particulars unknown to us, and perhaps placed beyond reach of direct examination. Nov. Org.,'

'Aph.,' c. iii., c. v.) Aristotle's definition coincides with the third, and induction " is an inference drawn from all particulars." (Prior Analy.,' ii., c. xxiii.) The second and third processes are improperly confounded by most writers on logic, and treated as one simple and

purely logical operation. But the second is not a logical process at all; since the conclusion is not necessarily inferrible from the premise, for the some of the antecedent does not necessarily legitimate the all of the conclusion, notwithstanding that the procedure may be warranted by the material problem of the science, or the fundamental principles of the human understanding. The third alone is properly an induction of logic; for logic does not consider things, but the general forms of thought under which the mind conceives them ; and the logical inference is not determined by any relation of causality between the premise and conclusion, but by the subjective relation of reason and consequence as involved in the thought. The inductive process is exactly the reverse of the deductive; for while the latter proceeds from the whole to the part, the former ascends from the part to the whole since it is only under the character of a constituted or containing whole, or as a constituent and contained part, that anything can become the term of logical argumentation. Of these two processes, Sir William Hamilton gives the following figures :Induction. Deduction.

:

XYZ are under B.
... XYZ are A.

or,

A contains B.

B contains XYZ.

A contains X Y Z.

This confusion of material and logical induc

tion led Gillies and others to insist on the sameness of the Baconian and Aristotelian induction; while Campbell and Dugald Stewart, who totally mistook the value of all logical inference, yet rightly maintained their difference.

By Aristotle, induction and deduction are viewed as in certain respects similar in form; but in others as diametrically opposed, the latter being an analysis of the whole into its parts, by descending from the more general to the more particular; but the former descends by a synthetical process from the parts to the whole. The logicians, who misapprehended the nature of induction, reduced it to a deductive syllogism of the third form, and thereby overthrew the validity of all deduction itself, since the latter is only possible by means of the former, which legitimates the proposition from which its reasoning proceeds.

Again, the Aristotelian induction was drawn from all the particulars, whereas the confusion which Sir W. Hamilton has pointed out gave rise to a division of the inductive process into perfect and imperfect, according as the enumeration of particulars is complete or incomplete. The latter gives only a probable result, whereas the necessity of the conclusion is essential to all logical inference, as its demonstrative stringency depends upon the form of the illation, and not upon the truth of the premises. It is proper to add, that no one ever knew the distinction between the imperfect and perfect forms of the conclusion better than Aristotle himself.

mathematics.

Induction (Mathematics). The method of induction, in the sense in which the word is used in natural philosophy, is not known in pure There certainly are instances in which a general proposition is proved by a collection of the demonstrations of different cases, which may remind the investigator of the inductive process, or the collection of the general from the particular. Such instances however must not be taken as permanent, for it usually happens that a general demonstration is discovered as soon as attention is turned to the subject.

There is however one particular method of proceeding which is extremely common in mathematical reasoning, and to which we propose to give the name of successive induction. It has the character of induction as defined by the logicians, because it is really the collection of a general truth from a demonstration which implies the examination of every particular case; but it adds to the necessary character of induction that each case depends upon one or more of those which precede. Substituting demonstration for observation, the mathematical process is truly inductive. A couple of instances of the method will enable the mathematical reader to recognize a mode of investigation with which he is already familiar,

Example 1.-The sum of any number of successive odd numbers, beginning from unity, is a square number, namely, the square of half the even number which follows the last odd number.