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 III. 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.

I.

2.

3.

Does the antecedent invariably produce an effect?
In what direction is that effect?

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 0'0000122 of its own length for every 1o Centigrade between 0° and 100o.

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

quantities of force in rubbing one substance against another, and showed that the heat produced was exactly greater or less in proportion as the force was greater or less. We can apply the method to many cases which had previously been treated by the simple method of difference; thus instead of striking a bell in a complete vacuum we can strike it with a very little air in the receiver of the air-pump, and we then hear a very faint sound, which increases or decreases every time we increase or decrease the density of the air. This experiment conclusively satisfies any person that air is the cause of the transmission of sound.

It is this method which often enables us to detect the material connection which exists between two bodies. For a long time it had been doubtful whether the red flames seen in total eclipses of the sun belonged to the sun or the moon; but during the last eclipse of the sun it was noticed that the flames moved with the sun, and were gradually covered and uncovered by the moon at successive instants of the eclipse. No one could doubt thenceforth that they belonged to the sun.

Whenever, again, phenomena go through Periodic Changes, alternately increasing and decreasing, we should seek for other phenomena which go through changes in exactly the same periods, and there will probably be a connection of cause and effect. It is thus that the tides are proved to be due to the attraction of the moon and sun, because the periods of high and low, spring and neap tides, succeed each other in intervals corresponding to the apparent revolutions of those bodies round the earth. The fact that the moon revolves upon its own axis in exactly the same period that it revolves round the earth, so that for unknown ages past the same side of the moon has always been turned towards the earth, is a most perfect case of concomitant variations, conclusively prov

ing that the earth's attraction governs the motions of the moon on its own axis.

The most extraordinary case of variations however consists in the connection which has of late years been shown to exist between the Aurora Borealis, magnetic storms, and the spots on the sun. It has only in the last 30 or 40 years become known that the magnetic compass needle is subject at intervals to very slight but curious movements; and that at the same time there are usually natural currents of electricity produced in telegraph-wires so as to interfere with the transmission of messages. These disturbances are known as magnetic storms, and are often observed to occur when a fine display of the Northern or Southern Lights is taking place in some part of the earth. Observations during many years have shown that these storms come to their worst at the end of every eleven years, the maximum taking place about the present year 1870, and then diminish in intensity until the next period of eleven years has passed. Close observations of the sun during 30 or 40 years have shown that the size and number of the dark spots, which are gigantic storms going on upon the sun's surface, increase and decrease exactly at the same periods of time as the magnetic storms upon the earth's surface. No one can doubt, then, that these strange phenomena are connected together, though the mode of the connection is quite unknown. It is now believed that the planets Jupiter, Saturn, Venus and Mars, are the real causes of the disturbances; for Balfour Stewart and Warren de la Rue have shown that an exact correspondence exists between the motions of these planets and the periods of the sunspots. This is a most remarkable and extensive case of concomitant variations.

We have now to consider a method of Induction which must be employed when several causes act at once