certain conclusion, but it asserts nothing beyond what was asserted in the premises. Mr Mill, indeed, differs from almost all other logicians in holding that Perfect Induction is improperly called Induction, because it does not lead to any new knowledge. He defines Induction as inference from the known to the unknown, and considers the unexamined cases which are apparently brought into our knowledge as the only gain from the process of reasoning. Hence Perfect Induction seems to him to be of no scientific value whatever, because the conclusion is a mere reassertion in a briefer form, a mere summing up of the premises. I may point out, however, that if Perfect Induction were no more than a process of abbreviation it is yet of great importance, and requires to be continually used in science and common life. Without it we could never make a comprehensive statement, but should be obliged to enumerate every particular. After examining the books in a library and finding them to be all English books we should be unable to sum up our results in the one proposition, "all the books in this library are English books;" but should be required to go over the list of books every time we desired to make any one acquainted with the contents of the library. The fact is, that the power of expressing a great number of particular facts in a very brief space is essential to the progress of science. Just as the whole science of arithmetic consists in nothing but a series of processes for abbreviating addition and subtraction, and enabling us to deal with a great number of units in a very short time, so Perfect Induction is absolutely necessary to enable us to deal with a great number of particular facts in a very brief space. It is usual to represent Perfect Induction in the form of an Inductive Syllogism, as in the following instance:Mercury, Venus, the Earth, &c., all move round the sun from West to East. Mercury, Venus, the Earth, &c., are all the known Planets. Therefore all the known planets move round the sun from West to East. This argument is a true Perfect Induction because the conclusion only makes an assertion of all known planets, which excludes all reference to possible future discoveries; and we may suppose that all the known planets have been enumerated in the premises. The form of the argument appears to be that of a syllogism in the third figure, namely Darapti, the middle term consisting in the group of the known planets. In reality, however, it is not an ordinary syllogism. The minor premise states not that Mercury, Venus, the Earth, Neptune, &c., are contained among the known planets, but that they are those planets, or are identical with them. This premise is then a doubly universal proposition of a kind (p. 184—7) not recognised in the Aristotelian Syllogism. Accordingly we may observe that the conclusion is a universal proposition, which is not allowable in the third figure of the syllogism. As another example of a Perfect Induction we may take January, February,............December, each contain less than 32 days. January......... December are all the months of the year. Therefore all the months of the year contain less than 32 days. Although Sir W. Hamilton has entirely rejected the notion, it seems worthy of inquiry whether the Inductive Syllogism be not really of the Disjunctive form of Syllogism. Thus I should be inclined to represent the last example in the form: A month of the year is either January, or February, or March............or December; but January has less than 32 days; and February has less than 32 days; and so on until we come to December, which has less than 32 days. It follows clearly that a month must in any case have less than 32 days; for there are only 12 possible cases, and in each case this is affirmed. The fact is that the major premise of the syllogism on the last page is a compound sentence with twelve subjects, and is therefore equivalent to twelve distinct logical propositions. The minor premise is either a disjunctive proposition, as I have represented it, or something quite different from anything we have elsewhere had. From Perfect Induction we shall have to pass to Imperfect Induction; but the opinions of Logicians are not in agreement as to the grounds upon which we are warranted in taking a part of the instances only, and concluding that what is true of those is true of all. Thus if we adopt the example found in many books and say— This, that, and the other magnet attract iron; we evidently employ a false minor premise, because this, that, and the other magnet which we have examined, cannot possibly be all existing magnets. In whatever form we put it there must be an assumption that the magnets which we have examined are a fair specimen of all magnets, so that what we find in some we may expect in all. Archbishop Whately considers that this assumption should be expressed in one of the premises, and he represents Induction as a Syllogism in Barbara as follows:That which belongs to this, that, and the other magnet, belongs to all; Attracting iron belongs to this, that, and the other; But though this is doubtless a correct expression of the assumption made in an Imperfect Induction, it does not in the least explain the grounds on which we are allowed to make the assumption, and under what circumstances such an assumption would be likely to prove true. Some writers have asserted that there is a Principle called the Uniformity of Nature, which enables us to affirm that what has often been found to be true of anything will continue to be found true of the same sort of thing. It must be observed, however, that if there be such a principle it is liable to exceptions; for many facts which have held true up to a certain point have afterwards been found not to be always true. Thus there was a wide and unbroken induction tending to show that all the Satellites in the planetary system went in one uniform direction round their planets. Nevertheless the Satellites of Uranus when discovered were found to move in a retrograde direction, or in an opposite direction to all Satellites previously known, and the same peculiarity attaches to the Satellite of Neptune more lately discovered. We may defer to the next lesson the question of the varying degree of certainty which belongs to induction in the several branches of knowledge. The advanced student may consult the following with advantage:-Mansel's Aldrich, Appendix, Notes G and H, Hamilton's Lectures on Logic, Lecture XVII., and Appendix VII., On Induction and Example, Vol. II., p. 358. J. S. Mill's System of Logic, Book III. Chap. 2, Of Inductions improperly so-called. LESSON XXVI. GEOMETRICAL AND MATHEMATICAL INDUCTION, ANALOGY AND EXAMPLE. IT is now indispensable that we should consider with great care upon what grounds Imperfect Induction is founded. No difficulty is encountered in Perfect Induction because all possible cases which can come under the general conclusion are enumerated in the premises, so that in fact there is no information in the conclusion which was not given in the premises. In this respect the Inductive Syllogism perfectly agrees with the general principles of deductive reasoning, which require that the information contained in the conclusion should be shown only from the data, and that we should merely unfold, or transform into an explicit statement what is contained in the premises implicitly. In Imperfect Induction the process seems to be of a wholly different character, since the instances concerning which we acquire knowledge may be infinitely more numerous than those from which we acquire the knowledge. Let us consider in the first place the process of Geometrical Reasoning which has a close resemblance to inductive reasoning. When in the fifth proposition of the first book of Euclid we prove that the angles at the base of an isosceles triangle are equal to each other, it is done by taking one particular triangle as an example. A figure is given which the reader is requested to regard as having two equal sides, and it is conclusively proved that if the sides be really equal then the angles opposite to those sides must be equal also. But Euclid says nothing about other isosceles triangles; he treats one single triangle as a sufficient specimen of all isosceles triangles, |