This is not an entirely fresh method, but may be regarded as a special case either of Agreement or of Difference, to prove the cause or effect, not of a phenomenon as a whole, but of some increment of it (positive or negative). There are certain forces, such as gravitation, heat, friction, that can never be eliminated altogether, and therefore can only be studied in their degrees. To such phenomena the method of Difference cannot be applied, because there are no negative instances.
But we may obtain negative instances of a given quant.i.ty of such a phenomenon (say, heat), and may apply the method of Difference to that quant.i.ty. Thus, if the heat of a body increases 10 degrees, from 60 to 70, the former temperature of 60 was a negative instance in respect of those 10 degrees; and if only one other circ.u.mstance (say, friction) has altered at the same time, that circ.u.mstance (if an antecedent) is the cause. Accordingly, if in the above Canon we insert, after "particular manner," "[no other change having concurred,]" it is a statement of the method of Difference as applicable to the increment of a phenomenon, instead of to the phenomenon as a whole; and we may then omit the last clause--"[or is connected, etc.]." For these words are inserted to provide for the case of co-effects of a common cause (such as the flash and report of a gun); but if no other change (such as the discharge of a gun) has concurred with the variations of two phenomena, there cannot have been a common cause, and they are therefore cause and effect.
If, on the other hand, we omit the clause "[no other change having concurred,]" the Canon is a statement of the method of Agreement as applicable to the increment of a phenomenon instead of to the phenomenon as a whole; and it is then subject to the imperfections of that method: that is to say, it leaves open the possibilities, that an inquirer may overlook a plurality of causes; or may mistake a connection of two phenomena, which (like the flash and report of a gun) are co-effects of a common cause, for a direct relation of cause and effect.
It may occur to the reader that we ought also to distinguish Qualitative and Quant.i.tative Variations as two orders of phenomena to which the present method is applicable. But, in fact, Qualitative Variations may be adequately dealt with by the foregoing methods of Agreement, Double Agreement, and Difference; because a change of quality or property entirely gets rid of the former phase of that quality, or subst.i.tutes one for another; as when the ptarmigan changes from brown to white in winter, or as when a stag grows and sheds its antlers with the course of the seasons. The peculiar use of the method of Variations, however, is to formulate the conditions of proof in respect of those causes or effects which cannot be entirely got rid of, but can be obtained only in greater or less amount; and such phenomena are or course, quant.i.tative.
Even when there are two parallel series of phenomena the one quant.i.tative and the other qualitative--like the rate of air-vibration and the pitch of sound, or the rate of ether-vibration and the colour-series of the spectrum--the method of Variations is not applicable. For (1) two such series cannot be said to vary together, since the qualitative variations are heterogeneous: 512: 576 is a definite ratio; but the corresponding notes, C, D, in the treble clef, present only a difference. Hence (2) the correspondence of each note with each number is a distinct fact. Each octave even is a distinct fact; there is a difference between C 64 and C 128 that could never have been antic.i.p.ated without the appropriate experience. There is, therefore, no such law of these parallel series as there is for temperature and change of volume (say) in mercury. Similar remarks apply to the physical and sensitive light-series.
We may ill.u.s.trate the two cases of the method thus (putting a dash against any letter, A" or _p_", to signify an increase or decrease of the phenomenon the letter stands for): Agreement in Variations (other changes being admissible)--
A B C A" D E A"" F G _p q r_ _p" s t_ _p"" u v_
Here the accompanying phenomena (_B C q r, D E s t, F G u v_) change from time to time, and the one thing in which the instances agree throughout is that any increase of A (A" or A"") is followed or accompanied by an increase of _p (p" or p"")_: whence it is argued that A is the cause of _p_, according to Prop. III. (a) (ch. xv. -- 7). Still, it is supposable that, in the second instance, D or E may be the cause of the increment of _p_; and that, in the third instance, F or G may be its cause: though the probability of such vicarious causation decreases rapidly with the increase of instances in which A and _p_ vary together.
And, since an actual investigation of this type must rely on observation, it is further possible that some undiscovered cause, X, is the real determinant of both A and _p_ and of their concomitant variations.
Professor Ferri, in his _Criminal Sociology_, observes: "I have shown that in France there is a manifest correspondence of increase and decrease between the number of homicides, a.s.saults and malicious wounding, and the more or less abundant vintage, especially in the years of extraordinary variations, whether of failure of the vintage (1853-5, 1859, 1867, 1873, 1878-80), attended by a remarkable diminution of crime (a.s.saults and wounding), or of abundant vintages (1850, 1856-8, 1862-3, 1865, 1868, 1874-5), attended by an increase of crime" (p. 117, Eng.
trans.). And earlier he had remarked that such crimes also "in their oscillations from month to month display a characteristic increase during the vintage periods, from June to December, notwithstanding the constant diminution of other offences" (p. 77). This is necessarily an appeal to the canon of Concomitant Variations, because France is never without her annual vintage, nor yet without her annual statistics of crime. Still, it is an argument whose cogency is only that of Agreement, showing that probably the abuse of the vintage is a cause of crimes of violence, but leaving open the supposition, that some other circ.u.mstance or circ.u.mstances, arising or varying from year to year, may determine the increase or decrease of crime; or that there is some unconsidered agent which affects both the vintage and crimes of violence. French sunshine, it might be urged, whilst it matures the generous grape, also excites a morbid fermentation in the human mind.
Difference in Variations may be symbolically represented thus (no other change having concurred):
A B A" B A"" B _p q_, _p" q_, _p"" q_.
Here the accompanying phenomena are always the same B/q; and the only point in which the successive instances differ is in the increments of A (A", A"") followed by corresponding increments of _p_ (_p", p""_): hence the increment of A is the cause of the increment of _p_.
For examples of the application of this method, the reader should refer to some work of exact science. He will find in Deschanel"s _Natural Philosophy_, c. 32, an account of some experiments by which the connection between heat and mechanical work has been established. It is there shown that "whenever work is performed by the agency of heat" [as in driving an engine], "an amount of heat disappears equivalent to the work performed; and whenever mechanical work is spent in generating heat" [as in rubbing two sticks together], "the heat generated is equivalent to the work thus spent." And an experiment of Joule"s is described, which consisted in fixing a rod with paddles in a vessel of water, and making it revolve and agitate the water by means of a string wound round the rod, pa.s.sed over a pulley and attached to a weight that was allowed to fall. The descent of the weight was measured by a graduated rule, and the rise of the water"s temperature by a thermometer. "It was found that the heat communicated to the water by the agitation amounted to one pound-degree Fahrenheit for every 772 foot-pounds of work" expended by the falling weight. As no other material change seems to take place during such an experiment, it shows that the progressive expenditure of mechanical energy is the cause of the progressive heating of the water.
The thermometer itself ill.u.s.trates this method. It has been found that the application of heat to mercury expands it according to a law; and hence the volume of the mercury, measured by a graduated index, is used to indicate the temperature of the air, water, animal body, etc., in which the thermometer is immersed, or with which it is brought into contact. In such cases, if no other change has taken place, the heat of the air, water, or body is the cause of the rise of the mercury in its tube. If some other substance (say spirit) be subst.i.tuted for mercury in constructing a thermometer, it serves the same purpose, provided the index be graduated according to the law of the expansion of that substance by heat, as experimentally determined.
Instances of phenomena that do not vary together indicate the exclusion of a supposed cause (by Prop. III (c)). The stature of the human race has been supposed to depend on temperature; but there is no correspondence. The "not varying together," however, must not be confused with "varying inversely," which when regular indicates a true concomitance. It is often a matter of convenience whether we regard concomitant phenomena as varying directly or inversely. It is usual to say--"the greater the friction the less the speed"; but it is really more intelligible to say--"the greater the friction the more rapidly molar is converted into molecular motion."
The Graphic Method exhibits Concomitant Variations to the eye, and is extensively used in physical and statistical inquiries. Along a horizontal line (the abscissa) is measured one of the conditions (or agents) with which the inquiry is concerned, called the Variable; and along perpendiculars (ordinates) is measured some phenomenon to be compared with it, called the Variant.
Thus, the expansion of a liquid by heat may be represented by measuring degrees of temperature along the horizontal, and the expansion of a column of the liquids in units of length along the perpendicular.
[Ill.u.s.tration: FIG. 9.]
In the next diagram (Fig. 10), reduced from one given by Mr. C.H. Denyer in an article on the Price of Tea (_Economic Journal_, No. 9), the condition measured horizontally is Time; and, vertically, three variants are measured simultaneously, so that their relations to one another from time to time may be seen at a glance. From this it is evident that, as the duty on tea falls, the price of tea falls, whilst the consumption of tea rises; and, in spite of some irregularity of correspondence in the courses of the three phenomena, their general causal connection can hardly be mistaken. However, the causal connection may also be inferred by general reasoning; the statistical Induction can be confirmed by a Deduction; thus ill.u.s.trating the combined method of proof to be discussed in the next chapter. Without such confirmation the proof by Concomitant Variations would not be complete; because, from the complexity of the circ.u.mstances, social statistics can only yield evidence according to the method of Agreement in Variations. For, besides the agents that are measured, there may always be some other important influence at work. During the last fifty years, for example, crime has decreased whilst education has increased: true, but at the same time wages have risen and many other things have happened.
[Ill.u.s.tration: FIG. 10.]
It will be noticed that in the diagram the three lines, especially those of Price and Consumption (which may be considered _natural_ resultants, in contrast with the arbitrary fixation of a Tax), do not depart widely from regular curves; and accordingly, a.s.suming the causes at work to vary continuously during the intervals between points of measurement, curves may be subst.i.tuted. In fact, a curve often represents the course of a phenomenon more truthfully than can be done by a line that zigzags along the exact measurements; because it is less influenced by temporary and extraordinary causes that may obscure the operation of those that are being investigated. On the other hand, the abrupt deviations of a punctilious zigzag may have their own logical value, as will appear in the next section.
In working with the Method of Variations one must allow for the occurrence in a series of "critical points," at which sudden and sometimes heterogeneous changes may take place. Every substance exists at different temperatures in three states, gaseous, liquid, solid; and when the change takes place, from one state to another, the series of variations is broken. Water, e.g., follows the general law that cooling is accompanied by decrease of volume between 212 and 39 F.: but above 212, undergoes a sudden expansion in becoming a gas; and below 39 begins to expand, until at 32 the expansion is considerable on its becoming solid. This ill.u.s.trates a common experience that concomitant variations are most regular in the "median range," and are apt to become irregular at the extremities of the series, where new conditions begin to operate.
The Canon of Variations, again, deals not with sudden irruptions of a cause, force or agent, but with some increase or decrease of an agent already present, and a corresponding increase or decrease of some other phenomenon--say an increase of tax and a rise of price. But there are cases in which the energy of a cause is not immediately discharged and dissipated. Whilst a tax of 6_d._ per lb. on tea raises the price per lb. by about 6_d._, however long it lasts, the continuous application of friction to a body may gradually raise its temperature to the point of combustion; because heat is received faster than it is radiated, and therefore acc.u.mulates. Such cases are treated by Mill under the t.i.tle of "progressive effects" (_Logic_: B. III., c. 15): he gives as an example of it the acceleration of falling bodies. The storage of effects is a fact of the utmost importance in all departments of nature, and is especially interesting in Biology and Sociology, where it is met with as heredity, experience, tradition. Evolution of species of plants and animals would (so far as we know) be impossible, if the changes (however caused) that adapt some individuals better than others to the conditions of life were not inherited by, and acc.u.mulated in, their posterity. The eyes in the peac.o.c.k"s tail are supposed to have reached their present perfection gradually, through various stages that may be ill.u.s.trated by the ocelli in the wings of the Argus pheasant and other genera of _Phasianidae_. Similarly the progress of societies would be impossible without tradition, whereby the improvements made in any generation may be pa.s.sed on to the next, and the experience of mankind may be gradually acc.u.mulated in various forms of culture. The earliest remains of culture are flint implements and weapons; in which we can trace the effect of tradition in the lives of our remote forefathers, as they slowly through thousands of years learnt to improve the chipping of flints, until the first rudely shaped lumps gave place to works of unmistakable design, and these to the beautiful weapons contemporary with the Bronze Age.
The Method of Gradations, the arranging of any phenomena to be studied in series, according to the degree in which some character is exhibited, is, perhaps, the most definite device in the Art of Discovery. (Bain: _Induction_, c. 6, and App. II.) If the causes are unknown it is likely to suggest hypotheses: and if the causes are partly known, variation in the character of the series is likely to indicate a corresponding variation of the conditions.
-- 5. THE CANON OF RESIDUES.
_Subduct from any phenomenon such part as previous inductions have shown to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents_.
The phenomenon is here a.s.sumed to be an effect: a similar Canon may be framed for residuary causes.
This also is not a fresh method, but a special case of the method of Difference. For if we suppose the phenomenon to be _p q r_, and the antecedent to be A B C, and that we already know B and C to have (either severally or together) the consequents _q r_, in which their efficacy is exhausted; we may regard
B C _q r_
as an instance of the absence of _p_ obtained deductively from the whole phenomenon
A B C _p q r_
by our knowledge of the laws of B and C; so that
A B C _p q r_
is an instance of the presence of _p_, differing otherwise from
B C _q r_
in nothing except that A is also present. By the Canon of Difference, therefore A is the cause of _p_. Or, again, when phenomena thus treated are strictly quant.i.tative, the method may be based on Prop. III. (b), ch. xv. -- 7.
Of course, if A can be obtained apart from B C and directly experimented with so as to produce _p_, so much the better; and this may often be done; but the special value of the method of Residues appears, when some complex phenomenon has been for the most part accounted for by known causes, whilst there remains some excess, or shortcoming, or deviation from the result which those causes alone would lead us to expect, and this residuary fact has to be explained in relation to the whole. Here the negative instance is const.i.tuted by deduction, showing what would happen but for the interference of some unknown cause which is to be investigated; and this prominence of the deductive process has led some writers to cla.s.s the method as deductive. But we have seen that all the Canons involve deduction; and, considering how much in every experiment is a.s.sumed as already known (what circ.u.mstances are "material," and when conditions may be called "the same"), the wonder is that no one has insisted upon regarding every method as concerned with residues. In fact, as scientific explanation progresses, the phenomena that may be considered as residuary become more numerous and the importance of this method increases.
Examples: The recorded dates of ancient eclipses having been found to differ from those a.s.signed by calculation, it appears that the average length of a day has in the meanwhile increased. This is a residuary phenomenon not accounted for by the causes formerly recognised as determining the rotation of the earth on its axis; and it may be explained by the consideration that the friction of the tides reduces the rate of the earth"s rotation, and thereby lengthens the day.
Astronomy abounds in examples of the method of Residues, of which the discovery of Neptune is the most famous.
Capillarity seems to be a striking exception to the principle that water (or any liquid) "finds its level," that being the condition of equilibrium; yet capillarity proves to be only a refined case of equilibrium when account is taken of the forces of adhesion exerted by different kinds of bodies in contact.
"Many of the new elements of Chemistry," says Herschel, "have been detected in the investigation of residual phenomena." Thus, Lord Rayleigh and Sir W. Ramsay found that nitrogen from the atmosphere was slightly heavier than nitrogen got from chemical sources; and, seeking the cause of this difference, discovered argon.
The Economist shows that when a country imports goods the chief means of paying for them is to export other goods. If this were all, imports and exports would be of equal value: yet the United Kingdom imports about 400,000,000 annually, and exports about 300,000,000. Here, then, is a residuary phenomenon of 100,000,000 to be accounted for. But foreign countries owe us about 50,000,000 for the use of shipping, and 70,000,000 as interest on the capital we have lent them, and 15,000,000 in commissions upon business transacted for them. These sums added together amount to 135,000,000; and that is 35,000,000 too much.
Thus another residuary phenomenon emerges; for whilst foreigners seem to owe us 435,000,000 they only send us 400,000,000 of imports. These 35,000,000 are accounted for by the annual investment of our capital abroad, in return for which no immediate payment is due; and, these being omitted, exports and imports balance. Since this was written the figures of our foreign trade have greatly risen; but the character of the explanation remains the same.
When, in pursuing the method of Variations, the phenomena compared do not always correspond in their fluctuations, the irregular movements of that phenomenon which we regard as the effect may often be explained by treating them as residuary phenomena, and then seeking for exceptional causes, whose temporary interference has obscured the influence of the general cause. Thus, returning to the diagram of the Price of Tea in -- 4, it is clear that generally the price falls as the duty falls; but in Mr. Denyer"s more minutely wrought diagram, from which this is reduced, it may be seen that in 1840 the price of tea rose from 3_s._ 9_d._ to 4_s._ 9_d._ without any increase of duty. This, however, is readily explained by the Chinese War of that year, which checked the supply.
Again, from 1869 to 1889 the duty was constant, whilst the price of tea fell as much as 8_d._ per lb.; but this residuary phenomenon is explained by the prodigiously increased production of tea during that period in India and Ceylon.
The above examples of the method of Residues are all quant.i.tative; but the method is often employed where exact estimates are un.o.btainable.
Thus Darwin, having found certain modifications of animals in form, coloration and habits, that were not clearly derivable from their struggle for existence in relation to other species or to external conditions, suggested that they were due to s.e.xual Selection.
The "vestiges" and "survivals" so common in Biology and Sociology are residuary phenomena. It is a general inference from the doctrine of Natural Selection that every organ of a plant, animal, or society is in some way useful to it. There occur, however, organs that have at present no a.s.signable utility, are at least wasteful, and sometimes even injurious. And the explanation is that formerly they were useful; but that, their uses having lapsed, they are now retained by the force of heredity or tradition. Either they are not injurious enough to be eliminated by natural selection; or they are correlated with other organs, whose utility outweighs their disutility.
CHAPTER XVII