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, 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 However often we try this it will be squares of a and b. = 7, the product of the sum and difference is 17 × 3 = 51; the squares of the quantities are 10 x 10 or 100 and 7 × 7 or 49, the difference of which is also 51. But however often we tried the rule no certainty would be added to it; because when proved algebraically there was no condition which restricted the result to any particular numbers, and a and might consequently be any numbers whatever. This generality of algebraical reasoning by which a property is proved of infinite varieties of numbers at once, is one of the chief advantages of algebra over arithmetic. There is also in algebra a process called Mathematical Induction or Demonstrative Induction, which shows the powers of reasoning in a very conspicuous way. A good example is found in the following problem :-If we take the first two consecutive odd numbers, I and 3, and add them together the sum is 4, or exactly twice two; if we take three such numbers 1+3+5, the sum is 9 or exactly three times three; if we take four, namely 1+3+5+7 the sum is 16, or exactly four times four; or generally, if we take any given number of the series, 1+3+5+7+... the sum is equal to the number of the terms multiplied by itself. Anyone who knows a very little algebra can prove that this remarkable law is universally true, as followsLet n be the number of terms, and assume for a moment that this law is true up to n terms, thus— I+3+5+7+......+(2n-1)=n2. Now add 2n+1 to each side of the equation. It follows that I+3+5+7+.. + (2n − 1) + (2n + 1) = n2+ 2n + 1. But the last quantity n2+2n + 1 is just equal to (n + 1)2; so that if the law is true for n terms it is true also for n + 1 terms. We are enabled to argue from each single case of the law to the next case; but we have already shown that it is true of the first few cases, therefore it must be true of all. By no conceivable labour could a person ascertain by trial what is the sum of the first billion odd numbers, and yet symbolically or by general reasoning we know with certainty that they would amount to a billion billion, and neither more nor less even by a unit. This process of Mathematical Induction is not exactly the same as Geometrical Induction, because each case depends upon the last, but the proof rests upon an equally narrow basis of experience, and creates knowledge of equal certainty and generality. Such mathematical truths depend upon observation of a few cases, but they acquire certainty from the perception we have of the exact similarity of one case to another, so that we undoubtingly believe what is true of one case to be true of another. It is very instructive to contrast with these cases certain other ones where there is a like ground of observation, but not the same tie of similarity. It was at one time believed that if any integral number were multipled by itself, added to itself and then added to 41, the result would be a prime number, that is a number which could not be divided by any other integral number except unity; in symbols, x2+x+41=prime number. This was believed solely on the ground of trial and experience, and it certainly holds for a great many values of x. Thus when x is successively made equal to the numbers in the first line below, the expression x2+x+41 gives the values in the second line, and they are all prime numbers: O I 2 3 4 5 6 7 8 9 ΙΟ 4I No reason however could be given why it should |