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. 133), 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 (p. 133). 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 By this process we should usually compare 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 two modes of proving a simple identity are so closely allied that it is doubtful how far we can detect any difference in their powers and instances of application. The first method is perhaps more convenient where the

Thus we prove

phenomena to be compared are rare. 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 rotate the plane of polarization of light are precisely those crystals which have 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 observe any of the four propositions A = AB, B= AB, a=ab, b=ab, all of which follow from A B (see p. 133).

=

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

Diamond = combustible gem.

In a similar manner we ascertain that

=

Mercury metal liquid at ordinary temperatures, Substance of least density = substance of least atomic weight.

Two or three objects may occasionally enter into the induction, as when we learn that

Sodium potassium metal of less density than

water,

Venus + Mercury Mars

=

major planet devoid of

satellites.

Induction of Partial Identities.

We found in the last section that the simple identity of two classes is almost always discovered not by direct observation of the fact, but by first establishing two partial identities. There are also a great multitude of cases in which the partial identity of one class with another is the only relation to be discovered. Thus the most common of all inductive inferences consists in establishing the fact that all objects having the properties of A have also those of B, or that A = AB. To ascertain the truth of a proposition of this kind it is merely necessary to assemble together, mentally or physically, all the objects included under A, and then observe whether B is present in each of them, or, which is the same, whether it would be impossible to select from among them any not-B. Thus, if we mentally assemble together all the heavenly bodies which move with apparent rapidity, that is to say the planets, we find that they all possess the property of not scintillating. We cannot analyse any vegetable substance without discovering that it contains carbon and hydrogen, but it is not true that all substances containing carbon and hydrogen are vegetable substances.

The great mass of scientific truths consists of propositions of this form A = AB. Thus in astronomy we learn that all the planets are spheroidal bodies; that they all revolve in one direction round the sun; that they all shine by reflected light; that they all obey the law of gravitation. But of course it is not to be asserted that all bodies obeying the law of gravitation, or shining by reflected light, or revolving in a particular direction, or being spheroidal in form, are planets. In other sciences we have immense numbers of propositions of the same form, as for instance that all substances in

But for the aggregate of alternatives we may now substitute their equivalent as given in the first premise, namely A, so that we get the required result

A = AX.

It may be remarked that we should have reached the same final result if our original premise had been of the form

A = AB+ AC+ ...... + AQ.

The difference of meaning is that all B's need not now be A's, nor all C's, &c. But we should still have

A = ABX + ACX+..... + AQX = AX.

We can always prove a proposition, if we find it more convenient, by proving its equivalent. To assert that all not-B's are not-A's, is exactly the same as to assert that all A's are B's. Accordingly we may ascertain that A = AB by first ascertaining that bab. If we observe, for instance, that all substances which are not solids are also not capable of double refraction, it follows necessarily that all double refracting substances are solids. We may convince ourselves that all electric substances are nonconductors of electricity, by reflecting that all good conductors do not, and in fact cannot, retain electric excitation. When we come to questions of probability it will be found desirable to prove, as far as possible, both the original proposition and its equivalent, as there is then an increased area of observation.

The number of alternatives which may arise in the division of a class varies greatly, and may be any number from two upwards. Thus it is probable that every substance is either magnetic or diamagnetic, and no substance can be both at the same time. The division then must be made in the form

A ABct AbС.

If now we can prove that all magnetic substances are capable of polarity, say BBC, and also that all