definite combinations, we are required to point out the laws which govern those combinations. Any laws being supposed, we can, with ease and certainty, decide whether the phenomena obey those laws. But the laws which may exist are infinite in variety, so that the chances are immensely against mere random guessing. The difficulty is much increased by the fact that several laws will usually be in operation at the same time, the effects of which are complicated together. The only modes of discovery consist either in exhaustively trying a great number of supposed laws, a process which is exhaustive in more senses than one, or else in carefully contemplating the effects, endeavouring to remember cases in which like effects followed from known laws. In whatever manner we accomplish the discovery, it must be done by the more or less conscious application of the direct process of deduction.

The Logical Alphabet illustrates induction as well as deduction. In considering the Indirect Process of Inference we found that from certain propositions we could infallibly determine the combinations of terms agreeing with those premises. The inductive problem is just the inverse. Having given certain combinations of terms, we need to ascertain the propositions with which the combinations are consistent, and from which they may have proceeded. Now, if the reader contemplates the following combinations,

he will probably remember at once that they belong to the premises A = AB, B = BC (p. 92). If not, he will require a few trials before he meets with the right answer, and every trial will consist in assuming certain laws and observing whether the deduced results agree with the data. To test the facility with which he can solve this inductive problem, let him casually strike out any of the combinations of the fourth column of the Logical Alphabet, (p. 94), and say what laws the remaining combinations obey, observing that every one of the letter-terms and their negatives ought to appear in order to avoid self-contradiction in the premises (pp. 74, III). Let him say, for instance, what laws are embodied in the combinations

The difficulty becomes much greater when more terms enter into the combinations. It would require some little examination to ascertain the complete conditions fulfilled in the combinations

The reader may discover easily enough that the principal laws are Ce, and A = Ae; but he would hardly discover without some trouble the remaining law, namely, that BD = BDe.

The difficulties encountered in the inductive investigations of nature, are of an exactly similar kind. We seldom observe any law in uninterrupted and undisguised operation. The acuteness of Aristotle and the ancient Greeks did not enable them to detect that all terrestrial bodies tend to fall towards the centre of the earth. A few nights of observation might have convinced an astronomer viewing the solar system from its centre, that the planets travelled round the sun; but the fact that our place of observation is one of the travelling planets, so complicates the apparent motions of the other bodies, that it required all the sagacity of Copernicus to prove the real simplicity of the planetary system. It is the same throughout nature; the laws may be simple, but their combined. effects are not simple, and we have no clue to guide us through their intricacies. "It is the glory of God," said Solomon, "to conceal a thing, but the glory of a king to search it out." The laws of nature are the invaluable secrets which God has hidden, and it is the kingly prerogative of the philosopher to search them out by industry and sagacity.

Inductive Problems for Solution by the Reader.

In the first edition (vol. ii. p. 370) I gave a logical problem involving six terms, and requested readers to discover the laws governing the combinations given. received satisfactory replies from readers both in the United States and in England. I formed the combina

I

tions deductively from four laws of correction, but my correspondents found that three simpler laws, equivalent to the four more complex ones, were the best answer; these laws are as follows: a = ac, b

=

cd, d

=

= Ef.

In case other readers should like to test their skill in the inductive or inverse problem, I give below several series of combinations forming problems of graduated difficulty.

Induction of Simple Identities.

Many important laws of nature are expressible in the forin of simple identities, and I can at once adduce them as examples to illustrate what I have said of the difficulty of the inverse process of induction. Two phenomena are conjoined. Thus all gravitating matter is exactly coincident with all matter possessing inertia; where one

property appears, the other likewise appears. All crystals of the cubical system, are all the crystals which do not doubly refract light. All exogenous plants are, with some exceptions, those which have two cotyledons or seed-leaves.

A little reflection will show that there is no direct and infallible process by which such complete coincidences may be discovered. Natural objects are aggregates of many qualities, and any one of those qualities may prove to be in close connection with some others. If each of a numerous group of objects is endowed with a hundred distinct physical or chemical qualities, there will be no less than (100 × 99) or 4950 pairs of qualities, which may be connected, and it will evidently be a matter of great intricacy and labour to ascertain exactly which qualities are connected by any simple law.

One principal source of difficulty is that the finite powers of the human mind are not sufficient to compare by a single act any large group of objects with another large group. We cannot hold in the conscious possession of the mind at any one moment more than five or six different ideas. Hence we must treat any more complex group by successive acts of attention. The reader will perceive by an almost individual act of comparison that the words Roma and Mora contain the same letters. He may perhaps see at a glance whether the same is true of Causal and Casual, and of Logica and Caligo. To assure himself that the letters in Astronomers make No more stars, that Serpens in akuleo is an anagram of Joannes Keplerus, or Great gun do us a sum an anagram of Augustus de Morgan, it will certainly be necessary to break up the act of comparison into several successive acts. The process will acquire a double character, and will consist in ascertaining that each letter of the first group is among the letters of the second group, and vice versa, that each letter of the second is among those of the first group. In the same way we can only prove that two long lists of names are identical, by showing that each name in one list occurs in the other, and vice versa.

This process of comparison really consists in establishing two partial identities, which are, as already shown (p. 58), equivalent in conjunction to one simple identity. We first ascertain the truth of the two propositions A = AB,

B = AB, and we then rise by substitution to the single law A = B.

=

There is another process, it is true, by which we may get to exactly the same result; for the two propositions A AB, a ab are also equivalent to the simple identity A = B. If then we can show that all objects included under A are included under B, and also that all objects not included under A are not included under B, our purpose is effected. By this process we should usually conpare two lists if we are allowed to mark them. For each name in the first list we should strike off one in the second, and if, when the first list is exhausted, the second list is also exhausted, it follows that all names absent from the first must be absent from the second, and the coincidence must be complete.

The

These two modes of proving an identity are so closely allied that it is doubtful how far we can detect any difference in their powers and instances of application. first method is perhaps more convenient when the phenomena to be compared are rare. Thus we prove that all the musical concords coincide with all the more simple numerical ratios, by showing that each concord arises from a simple ratio of undulations, and then showing that each simple ratio gives rise to one of the concords. To examine all the possible cases of discord or complex ratio of undulation would be impossible. By a happy stroke of induction Sir John Herschel discovered that all crystals of quartz which cause the plane of polarization of light to rotate are precisely those crystals which Lave plagihedral faces, that is, oblique faces on the corners of the prism unsymmetrical with the ordinary faces. This singular relation would be proved by observing that all plagihedral crystals possessed the power of rotation, and vice versa all crystals possessing this power were plagihedral. But it might at the same time be noticed that all ordinary crystals were devoid of the power. There is no reason why we should not detect any of the four propositions A AB, a = ab, b ab, all of which follow from A B (p. 115).

=

AB, B

=

=

=

Sometimes the terms of the identity may be singular objects; thus we observe that diamond is a combustible gem, and being unable to discover any other that is, we affirm--

K