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qualities of the same substance in the solid and liquid condition without any other change of circumstances, because it is necessary to alter the temperature of the substance in order to liquefy or solidify it. The temperature might thus be the cause of what we attribute to the liquid or solid condition. Under such circumstances we have to resort to what Mr Mill calls the joint method of agreement and difference, which consists in a double application of the method of agreement, first to a number of instances where an effect is produced, and secondly, to a number of quite different instances where the effect is not produced. It is clearly to be understood, however, that the negative instances differ in several circumstances from the positive ones; for if they differed only in one circumstance we might apply the simple method of differ

Iceland spar, for instance, has a curious power of rendering things seen through it apparently double. This phenomenon, called double refraction, also belongs to many other crystals; and we might at once prove it to be due to crystalline structure could we obtain any transparent substance crystallized and uncrystallized, but subject to no other alteration. We have, however, a pretty satisfactory proof by observing that uniform transparent uncrystallized substances agree in not possessing double refraction, and that crystalline substances, on the other hand, with certain exceptions which are easily explained, agree in possessing the power in question. The principle of the joint method may be stated in the following rule, which is Mr Mill's Third Canon :

“If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance; the circumstance in which alone the two sets of instances (always or invariably) differ, is the effect, or the cause,

or an indispensable part of the cause, of the phenomenon.”

I have inserted the words in parentheses, as without them the canon seems to me to express exactly the opposite of what Mr Mill intends.

It may facilitate the exact comprehension of these inductive methods if I give the following symbolic representation of them in the manner adopted by Mr Mill. Let A, B, C, D, E, &c., be antecedents which may be variously combined, and let a, b, c, d, e, &c., be effects following from them. If then we can collect the following sets of antecedents and effectsAntecedents.



ade AFG

afg АНК


we may apply the method of agreement, and little doubt
will remain that A, the sole invariable antecedent, is the
cause of a.
The method of difference is sufficiently represented by-



bc Here while B and C remain perfectly unaltered we find that the presence or absence of A occasions the presence or absence of a, of which it is therefore the cause, in the presence of B and C. But the reader may be cautioned against thinking that this proves A to be the cause of a under all circumstances whatever.

The joint method of agreement and difference is similarly represented by

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Here the presence of A is followed as in the simple method of agreement by a; and the absence of A, in circumstances differing from the previous ones, is followed by the absence of a. Hence there is a very high probability that A is the cause of a. But it will easily be seen that A is not the only circumstance in which the two sets of instances differ, otherwise to any pair we might apply the simple method of difference. But the presence of A is a circumstance in which one set invariably, or uniformly, or always, differs, from the other set. This joint method is thus a substitute for the simpler method of difference in cases where that cannot be properly brought into action.

Herschel's Discourse, part II. chap. 6, p. 144.
Mill's System of Logic, book 111. chaps. 8 and 9.

LESSON XXIX. METHODS OF QUANTITATIVE INDUCTION. The methods of Induction described in the last Lesson related merely to the happening or not happening of the event, the cause of which was sought. Thus we learnt that friction was one cause of heat by observing that two



solid bodies, even two pieces of ice, rubbed together, produced heat, but that when they were not rubbed there was no such production of heat. This, however, is a very elementary sort of experiment; and in the progress of an investigation we always require to measure the exact quantity of an effect, if it be capable of being more or less, and connecting it with the quantity of the cause. There is in fact a natural course of progress through which we proceed in every such inquiry, as may be stated in the following series of questions.

Does the antecedent invariably produce an effect?

In what direction is that effect? 3. How much is that effect in proportion to the cause? 4. Is it uniformly in that proportion? 5. If not, according to what law does it vary?

Take for instance the effect of heat in altering the dimensions of bodies. The first question is, whether the heating of a solid body, say a bar of iron, alters its length; the simple method of difference enables us to answer that it does. The next inquiry shows that almost all substances are lengthened or increased in dimensions by heat, but that a very few, such as india rubber, and water below 4'080 Cent., are decreased. We next ascertain the proportion of the change to each degree of temperature, which is called the coefficient of expansion. Thus iron expands o‘oo00122 of its own length for every 1o Centigrade between 0 and 100°.

Still more minute inquiry shows, however, that the expansion is not uniformly proportional to temperature; most metals expand more and more rapidly the hotter they are, but the details of the subject need not be considered here.

The fixed stars, again, have often been mentioned in these Lessons, but the reader is probably aware that they are not really fixed. Taking any particular star, the

astronomer has really to answer the several five questions stated below.

Firstly. Does the star move?
2ndly. In what direction does it move?
3rdly. How much does it move in a year or a century?
4thly. Does it move uniformly?

5thly. If not, according to what law does the motion vary in direction and rapidity?

Every science and every question in science is first a matter of 'fact only, then a matter of quantity, and by degrees becomes more and more precisely quantitative. Thirty years ago most of the phenomena of electricity and electro-magnetism were known merely as facts; now they can be for the most part exactly measured and calculated.

As soon as phenomena can thus be measured we can apply a further Method of Induction of a very important character. It is the Method of Difference indeed applied under far more favourable circumstances, where every degree and quantity of a phenomenon gives us a new experiment and proof of connection between cause and effect. It may be called the Method of Concomitant Variations, and is thus stated by Mr Mill, in what he entitles the Fifth Canon of Induction :

“Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation."

Sir John Herschel's statement of the same method is as follows:"Increase or diminution of the effect, with the increased or diminished intensity of the cause, in cases which admit of increase and diminution,” to which he adds, "Reversal of the effect with that of the cause."

The illustrations of this method are infinitely numerous. Thus Mr Joule, of Manchester, conclusively proved that friction is a cause of heat by expending exact

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