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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. 11., P. 358. J. S. Mill's System of Logic, Book 111. Chap. 2, Of Inductions improperly so-called.
GEOMETRICAL AND MATHEMATICAL INDUC.
TION, 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,
and we are asked to believe that what is true of that is true of any other, whether its sides be so small as to be only visible in a microscope, or so large as to reach to the furthest fixed star. There may evidently be an infinite number of isosceles triangles as regards the length of the equal sides, and each of these may be infinitely varied by increasing or diminishing the contained angle, so that the number of possible isosceles triangles is infinitely infinite; and yet we are asked to believe of this incomprehensible number of objects what we have proved only of one single specimen. This might seem to be the most extremely Imperfect Induction possible, and yet every one allows that it gives us really certain knowledge. We do know with as much certainty as knowledge can possess, that if lines be conceived as drawn from the earth to two stars equally distant, they will make equal angles with the line joining those stars; and yet we can never have tried the experiment.
The generality of this geometrical reasoning evidently depends upon the certainty with which we know that all isosceles triangles exactly resemble each other. The proposition proved does not in fact apply to a triangle unless it agrees with our specimen in all the qualities essential to the proof. The absolute length of any of the sides or the absolute magnitude of the angle contained between any of them were not points upon which the proof depended—they were purely accidental circumstances; hence we are at perfect liberty to apply to all new cases of an isosceles triangle what we learn of one case. Upon a similar ground rests all the vast body of certain knowledge contained in the mathematical sciences—not only all the geometrical truths, but all general algebraical truths. It was shown, for instance, in p. 58, that if a and b be two quantities, and we multiply together their sum and difference, we get the difference of the
the order of nature in outward things or in the mind; more, he can neither know nor do."
The above is the first of the aphorisms or paragraphs with which the Novum Organum commences. In the second aphorism he asserts that the unaided mind can effect little and is liable to err; assistance in the form of a definite logical method is requisite, and this it was the purpose of his New Instrument to furnish. The 3rd and 4th aphorisms must be given entire; they are:
“Human science and human power coincide, because ignorance of a cause deprives us of the effect. For nature is not conquered except by obedience; and what we discover as a cause by contemplation becomes a rule in operation."
“Man can himself do nothing else than move natural bodies to or from each other; nature working within accomplishes the rest."
It would be impossible more clearly and completely to express the way in which we discover science by interpreting the changes we obserye in nature, and then turn our knowledge to a useful purpose in the promotion of the arts and manufactures. We cannot create and we cannot destroy a particle of matter; it is now known that we cannot even create or destroy, force; nor can we really alter the inner nature of any substance that we have to deal with. All that we can do is to observe carefully how one substance by its natural powers acts upon another substance, and then by rioving them together at the right time we can effect our object; as Bacon says, “Nature working within does the rest.” Had it not been the nature of heat when applied to water to develope steam possessing elastic power, it is needless to say that the steam-engine could never have been made, so that the invention of the steam-engine arose from observing the utility of the force of steam, and employing it accordingly,
It is in this sense that Virgil has proclaimed him happy who knows the causes of things
Felix qui potuit rerum cognoscere causas, and that Bacon has said, Knowledge is Power. So far as we have observed how things happen in nature, and on what occasion particular effects are brought to pass, we are enabled to avoid or utilise those effects. as we may desire, not by altering the natures of things, but by allowing them in suitable times and circumstances to manifest their own proper powers. It is thus, as Tennyson has excellently said, that we
“Rule by obeying Nature's Powers." Inductive logic treats of the methods of reasoning by which we may successfully interpret nature and learn the natural laws which various substances obey in different circumstances. In this lesson we consider the first requisite of induction, namely, the experience or examination of nature which is requisite to furnish us with facts. Such experience is obtained either by observation or experiment. To observe is merely to notice events and changes which are produced in the ordinary course of nature, without being able, or at least attempting, to control or vary those changes. Thus the early astronomers, observed the motions of the sun, moon and planets among the fixed stars, and gradually detected many of the laws or periodical returns of those bodies. Thus it is that the meteorologist observes the ever-changing weather, and notes the height of the barometer, the temperature and moistness of the air, the direction and force of the wind, the height and character of the clouds, without being in the least able to govern any of these facts. The geologist again is genenerally a simple observer when he investigates the nature and position of rocks. The zoologist, the botanist, and