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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 x+x+41 gives the values in the second line, and they are all prime numbers:

0 1 2 3 4 5 6 7 8 9 10 41 43 47 53 61 71 83 97 113 131 151

No reason however could be given why it should always be true, and accordingly it was found that the rule does not always hold true, but fails when x=40. Then we have 40X40 +40+41=1681, but this is clearly equal to 41 × 40+41 or 41 × 41, and is not a prime number.

In that branch of mathematics which treats of the peculiar properties and kinds of numbers, other propositions depending solely upon observation have been asserted to be always true.

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Thus Fermat believed that 22+1 always represents a prime number, but could not give any reason for the assertion. It holds true in fact until the product reaches the large number 4294967297, which was found to be divisible by 641, so that the generality of the statement was disproved.

We find then that in some cases a single instance proves a general and certain rule, while in others a very great number of instances are insufficient to give any certainty at all; all de

pends upon the perception we have of similarity or identity between one case and another. We can perceive no similarity between all prime numbers which assures us that because one is represented by a certain formula, also another is; but we do find such similarity between the sums of odd numbers, or between isosceles triangles.

Exactly similar considerations apply to inductions in physical science. When a chemist analyses a few grains of water and finds that they contain exactly 8 parts of oxygen and 1 of hydrogen for 9 parts of water, he feels warranted in asserting that the same is true of all pure water whatever be its origin, and whatever be the part of the world from which it comes. But if he analyse a piece of granite, or a sample of seawater from one part of the world, he does not feel any confidence that it will resemble exactly a piece of granite, or a sample of sea-water from another part of the earth; hence he does not venture to assert of all granite or sea-water, what he finds true of a single sample. Extended experience shows that granite is very variable in composition, but that sea-water is rendered pretty uniform by constant mixture of currents. Nothing but experience in these cases could inform us how far we may assert safely of one sample what we have ascertained of another. But we have reason to believe that chemical compounds are naturally fixed and invariable in composition, according to Dalton's laws of combining proportions. No à priori reasoning from

the principles of thought could have told us this, and we only learn it by extended experiment. But having once shown it to be true with certain substances we do not need to repeat the trial with all other substances, because we have every reason to believe that it is a natural law in which all chemical substances resemble each other. It is only necessary then for a single accurate analysis of a given fixed compound to be made in order to inform us of the composition of all other portions of the same substance.

It must be carefully observed however that all inductions in physical science are only probable, or that if certain, it is only hypothetical certainty they possess. Can I be absolutely certain that all water contains one part of hydrogen in nine? I am certain only on two conditions:

1. That this was certainly the composition of the sample tried.

2. That any

other substance I call water ex

actly resembles that sample.

But even if the first condition be undoubtedly true, I cannot be certain of the second. For how do I know what is water except by the fact of its being a transparent liquid, freezing into a solid and evaporating into steam, possessing a high specific heat, and a number of other distinct properties? But can I be absolutely certain that every liquid possessing all these properties is water? Practically I can be certain, but theoretically I cannot.

7. From Mill's System of Logic, pp. 125, 126: 210-228, 8vo edition.

Reasoning, in the extended sense in which I use the term, and in which it is synonymous with Inference, is popularly said to be of two kinds reasoning from particulars to generals, and reasoning from generals to particulars; the former being called Induction, the latter Ratiocination or Syllogism. It will presently be shown that there is a third species of reasoning, which falls under neither of these descriptions, and which, nevertheless, is not only valid, but is the foundation of both the others. Of Induction, therefore, we shall say no more at present, than that it at least is, without doubt, a process of real inference. The conclusion in an induction embraces more than is contained in the premises. The principle or law collected from particular instances, the general proposition in which we embody the result of our experience, covers a much larger extent of ground than the individual experiments which form its basis. A principle ascertained by experience, is more than a mere summing up of what has been specifically observed in the individual cases which have been examined; it is a generalization grounded on those cases, and expressive of our belief, that what we there found true is true in an indefinite number of cases which we have not examined, and are never likely to examine. The nature and grounds of this inference, and the conditions necessary to make it legitimate, will be the subject of discus

sion in the Third Book: but that such inference really takes place is not susceptible of question. In every induction we proceed from truths which we knew, to truths which we did not know; from facts certified by observation, to facts which we have not observed, and even to facts not capable of being now observed; future facts, for example; but which we do not hesitate to believe on the sole evidence of the induction itself. Induction, then, is a real process of Reasoning or Inference.

OF INDUCTIONS IMPROPERLY SO CALLED.

§ 1. Induction, then, is 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, 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 in similar circumstances at all times.

This definition excludes from the meaning of the term Induction, various logical operations, to which it is not unusual to apply that name.

Induction, as above defined, is a process of inference; it proceeds from the known to the unknown; and any operation involving no inference, any process in which what seems the conclusion is no wider than the premises from which it is drawn, does not fall within the meaning of the term. Yet in the common books of Logic we find this laid down as the most perfect, in

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