Logic Part 20

The comprehension of the quant.i.tative aspect of causation is greatly aided by Bain's a.n.a.lysis of any cause into a 'Moving or an Inciting Power' and a 'Collocation' of circ.u.mstances. When a demagogue by making a speech stirs up a mob to a riot, the speech is the moving or inciting power; the mob already in a state of smouldering pa.s.sion, and a street convenient to be wrecked, are the collocation. When a small quant.i.ty of strychnine kills a man, the strychnine is the inciting power; the nature of his nervo-muscular system, apt to be thrown into spasms by that drug, and all the organs of his body dependent on that system, are the collocation. Now any one who thinks only of the speech, or the drug, in these cases, may express astonishment at the disproportion of cause and effect:

"What great events from trivial causes spring!"

But, remembering that the whole cause of the riot included the excited mob, every one sees that its muscular power is enough to wreck a street; and remembering that breathing depends upon the normal action of the intercostal muscles, it is plain that if this action is stopped by strychnine, a man must die. Again, a slight rise of temperature may be a sufficient inciting power to occasion extensive chemical changes in a collocation of elements otherwise stable; a spark is enough to explode a powder magazine. Hence, when sufficient energy to account for any effect cannot be found in the inciting power, or manifestly active condition, we must look for it in the collocation which is often supposed to be pa.s.sive.

And that reminds us of another common misapprehension, namely, that in Nature some things are pa.s.sive and others active: the distinction between 'agent' and 'patient.' This is a merely relative distinction: in Nature all things are active. To the eye some things seem at rest and others in motion; but we know that nothing is really at rest, that everything palpitates with molecular change, and whirls with the planet through s.p.a.ce. Everything that is acted upon reacts according to its own nature: the quietest-looking object (say, a moss-covered stone), if we try to push or lift it, pushes or pulls us back, a.s.suring us that 'action and reaction are equal and opposite.' 'Inertia' does not mean want of vigour, but may be metaphorically described as the inexpugnable resolve of everything to have its own way.

The equality of cause and effect defines and interprets the unconditionality of causation. The cause, we have seen, is that group of conditions which, without any further condition, is followed by a given event. But how is such a group to be conceived? Unquantified, it admits only of a general description: quantified, it must mean a group of conditions equal to the effect in ma.s.s and energy, the essence of the physical world. Apparently, a necessary conception of the human mind: for if a cause seem greater than its effect, we ask what has become of the surplus matter and energy; or if an effect seem greater than its cause, we ask whence the surplus matter and energy has arisen. So convinced of this truth is every experimenter, that if his results present any deviation from it, he always a.s.sumes that it is he who has made some mistake or oversight, never that there is indeterminism or discontinuity in Nature.

The transformation of matter and energy, then, is the essence of causation: because it is continuous, causation is immediate; and because in the same circ.u.mstances the transformation always follows the same course, a cause has invariably the same effect. If a fire be lit morning after morning in the same grate, with coal, wood, and paper of the same quality and similarly arranged, there will be each day the same flaming of paper, crackling of wood and glowing of coal, followed in about the same time by the same reduction of the whole ma.s.s partly to ashes and partly to gases and smoke that have gone up the chimney. The flaming, crackling and glowing are, physically, modes of energy; and the change of materials into gas and ashes is a chemical and physical redistribution: and, if some one be present, he will be aware of all this; and then, besides the physical changes, there will be sensations of light, sound and heat; and these again will be always the same in the same circ.u.mstances.

The Cause of any event, then, when exactly ascertainable, has five marks: it is (quant.i.tatively) _equal_ to the effect, and (qualitatively) _the immediate, unconditional, invariable antecedent of the effect_.

-- 3. This scientific conception of causation has been developed and rendered definite by the investigations of those physical sciences that can avail themselves of exact experiments and mathematical calculation; and it is there, in Chemistry and Physics, that it is most at home. The conception can indeed be carried into the Biological and Social Sciences, even in its quant.i.tative form, by making the proper allowances. For the limbs of animals are levers, and act upon mechanical principles; and digestion and the aeration of the blood by breathing are partly chemical processes. There is a quant.i.tative relation between the food a man eats and the amount of work he can do. The numbers of any species of plant or animal depend upon the food supply. The value of a country's imports is equal to the value of its exports and of the services it renders to foreigners. But, generally, the less experiment and exact calculation are practicable in any branch of inquiry, the less rigorously can the conception of causation be applied there, the more will its application depend upon the qualitative marks, and the more need there will be to use it judiciously. In every inquiry the greatest possible precision must be aimed at; but it is unreasonable to expect in any case more precise proof than the subject admits of in the existing state of culture.

Wherever mental action is involved, there is a special difficulty in applying the physical notion of causation. For if a Cause be conceived of as matter in motion, a thought, or feeling, or volition can be neither cause nor effect. And since mental action is involved in all social affairs, and in the life of all men and animals, it may seem impossible to interpret social or vital changes according to laws of causation. Still, animals and men are moving bodies; and it is recognised that their thoughts and feelings are so connected with their movements and with the movements of other things acting upon them, that we can judge of one case by another; although the connection is by no means well understood, and the best words (such as all can agree to use) have not yet been found to express even what we know about it. Hence, a regular connection being granted, I have not hesitated, to use biological and social events and the laws of them, to ill.u.s.trate causation and induction; because, though less exact than chemical or mechanical examples, they are to most people more familiar and interesting.

In practical affairs, it is felt that everything depends upon causation; how to play the fiddle, or sail a yacht, or get one's living, or defeat the enemy. The price of pig-iron six months hence, the prospects of the harvest, the issue in a Coroner's Court, Home Rule and Socialism, are all questions of causation. But, in such cases, the conception of a cause is rarely applied in its full scientific acceptation, as the unconditional antecedent, or 'all the conditions' (neither more nor less) upon which the event depends. This is not because men of affairs are bad logicians, or incapable of scientific comprehension; for very often the reverse is conspicuously true; but because practical affairs call for prompt.i.tude and a decisive seizing upon what is predominantly important. How learn to play the fiddle? "Go to a good teacher." (Then, beginning young enough, with natural apt.i.tude and great diligence, all may be well.) How defeat the enemy? "Be two to one at the critical juncture." (Then, if the men are brave, disciplined, well armed and well fed, there is a good chance of victory.) Will the price of iron improve?

"Yes: for the market is oversold": (that is, many have sold iron who have none to deliver, and must at some time buy it back; and that will put up the price--if the stock is not too great, if the demand does not fall off, and if those who have bought what they cannot pay for are not in the meanwhile obliged to sell.) These prompt and decisive judgments (with the parenthetic considerations unexpressed) as to what is the Cause, or predominantly important condition, of any event, are not as good as a scientific estimate of all the conditions, when this can be obtained; but, when time is short, the insight of trained sagacity may be much better than an imperfect theoretical treatment of such problems.

-- 4. To regard the Effect of certain antecedents in a narrow selective way, is another common mistake. In the full scientific conception of an Effect it is the sum of the unconditional consequences of a given state and process of things: the consequences immediately flowing from that situation without further conditions. Always to take account of all the consequences of any cause would no doubt be impracticable; still the practical, as well as the scientific interest, often requires that we should enlarge our views of them; and there is no commoner error in private effort or in legislation than to aim at some obvious good, whilst overlooking other consequences of our action, the evil of which may far outweigh that good. An important consequence of eating is to satisfy hunger, and this is the ordinary motive to eat; but it is a poor account of the physiological consequences. An important consequence of firing a gun is the propulsion of the bullet or sh.e.l.l; but there are many other consequences in the whole effect, and one of them is the heating of the barrel, which, acc.u.mulating with rapid firing, may at last put the gun out of action. The tides have consequences to s.h.i.+pping and in the wear and tear of the coast that draw every one's attention; but we are told that they also r.e.t.a.r.d the rotation of the earth, and at last may cause it to present always the same face to the sun, and, therefore, to be uninhabitable. Such concurrent consequences of any cause may be called its Co-effects: the Effect being the sum of them.

The neglect to take account of the whole effect (that is, of all the co-effects) in any case of causation is perhaps the reason why many philosophers have maintained the doctrine of a "Plurality of Causes": meaning not that more than one condition is operative in the antecedent of every event (which is true), but that the same event may be due at different times to different antecedents, that in fact there may be _vicarious_ causes. If, however, we take any effect as a whole, this does not seem to be true. A fire may certainly be lit in many ways: with a match or a flint and steel, or by rubbing sticks together, or by a flash of lightning: have we not here a plurality of causes? Not if we take account of the whole effect; for then we shall find it modified in each case according to the difference of the cause. In one case there will be a burnt match, in another a warm flint, in the last a changed state of electrical tension. And similar differences are found in cases of death under different conditions, as stabbing, hanging, cholera; or of s.h.i.+pwreck from explosion, scuttling, tempest. Hence a Coroner's Court expects to find, by examining a corpse, the precise cause of death. In short, if we knew the facts minutely enough, it would be found that there is only one Cause (sum of conditions) for each Effect (sum of co-effects), and that the order of events is as uniform backwards as forwards.

Still, as we are far from knowing events minutely, it is necessary in practical affairs, and even in the more complex and unmanageable scientific investigations, especially those that deal with human life, to acknowledge a possible plurality of causes for any effect. Indeed, forgetfulness of this leads to many rash generalisations; as that 'revolutions always begin in hunger'; or that 'myths are a disease of language.' Then there is great waste of ingenuity in reconciling such propositions with the recalcitrant facts. A scientific method recognises that there may be other causes of effects thus vaguely conceived, and then proceeds to distinguish in each cla.s.s of effects the peculiarities due to different causes.

-- 5. The understanding of the complex nature of Causes and Effects helps us to overcome some other difficulties that perplex the use of these words. We have seen that the true cause is an _immediate_ antecedent; but if the cause is confounded with _one_ of its const.i.tuent conditions, it may seem to have long preceded the event which is regarded as its effect. Thus, if one man's death is ascribed to another's desire of revenge, this desire may have been entertained for years before the a.s.sa.s.sination occurred: similarly, if a s.h.i.+pwreck is ascribed to a sunken reef, the rock was waiting for ages before the s.h.i.+p sailed that way. But, of course, neither the desire of revenge nor the sunken rock was 'the sum of the conditions' on which the one or the other event depended: as soon as this is complete the effect appears.

We have also seen the true effect of any state and process of things is the immediate consequence; but if the effect be confounded with _one_ of its const.i.tuent factors, it may seem to long outlive the cessation of the cause. Thus, in nearly every process of human industry and art, one factor of the effect--a road, a house, a tool, a picture--may, and generally does, remain long after the work has ceased: but such a result is not the whole effect of the operations that produce it. The other factors may be, and some always are, evanescent. In most of such works some heat is produced by hammering or friction, and the labourers are fatigued; but these consequences soon pa.s.s off. Hence the effect as a whole only momentarily survives the cause. Consider a pendulum which, having been once set agoing, swings to and fro in an arc, under the joint control of the shaft, gravitation and its own inertia: at every moment its speed and direction change; and each change may be considered as an effect, of which the antecedent change was one condition. In such a case as this, which, though a very simple, is a perfectly fair example of all causation, the duration of either cause or effect is quite insensible: so that, as Dr. Venn says, an Effect, rigorously conceived, is only "the initial tendency" of its Cause.

-- 6. Mill contrasted two forms under which causation appears to us: that is to say, the conditions const.i.tuting a cause may be modified, or 'intermixed' in the effect, in two ways, which are typified respectively by Mechanical and Chemical action. In mechanical causation, which is found in Astronomy and all branches of Physics, the effects are all reducible to modes of energy, and are therefore commensurable with their causes. They are either directly commensurable, as in the cases treated of in the consideration of the mechanical powers; or, if different forms of energy enter into cause and effect, such as mechanical energy, electrical energy, heat, these different forms are severally reducible to units, between which equivalents have been established. Hence Mill calls this the "h.o.m.ogeneous intermixture of effects," because the antecedents and consequents are fundamentally of the same kind.

In chemical causation, on the other hand, cause and effect (at least, as they present themselves to us) differ in almost every way: in the act of combination the properties of elements (except weight) disappear, and are superseded by others in the compound. If, for example, mercury (a heavy, silvery liquid) be heated in contact with oxygen (a colourless gas), oxide of mercury is formed (red precipitate, which is a powder).

This compound presents very different phenomena from those of its elements; and hence Mill called this cla.s.s of cases "the heteropathic intermixture of effects." Still, in chemical action, the effect is not (in Nature) heterogeneous with the cause: for the weight of a compound is equal to the sum of the weights of the elements that are merged in it; and an equivalence has been ascertained between the energy of chemical combination and the heat, light, etc., produced in the act of combination.

The heteropathic intermixture of effects is also found in organic processes (which, indeed, are partly chemical): as when a man eats bread and milk, and by digestion and a.s.similation converts them into nerve, muscle and bone. Such phenomena may make us wonder that people should ever have believed that 'effects resemble their causes,' or that 'like produces like.' A dim recognition of the equivalence of cause and effect in respect of matter and motion may have aided the belief; and the resemblance of offspring to parents may have helped: but it is probably a residuum of magical rites; in which to whistle may be regarded as a means of raising the wind, because the wind whistles; and rain-wizards may make a victim shed tears that the clouds also may weep.

-- 7. Another consideration arises out of the complex character of causes and effects. When a cause consists of two or more conditions or forces, we may consider what effect any one of them would have if it operated alone, that is to say, its _Tendency_. This is best ill.u.s.trated by the Parallelogram of Forces: if two forces acting upon a point, but not in the same direction, be represented by straight lines drawn in the direction of the forces, and in length proportional to their magnitudes, these lines, meeting in an angle, represent severally the tendencies of the forces; whilst if the parallelogram be completed on these lines, the diagonal drawn from the point in which they meet represents their _Resultant_ or effect.

Again, considering the tendency of any force if it operated alone, we may say that, when combined with another force (not in the same direction) in any resultant, its tendency is _counteracted_: either partially, when the direction of the resultant is different; or wholly when, the other force being equal and opposite, the resultant is equilibrium. If the two forces be in the same direction, they are merely added together. Counteraction is only one mode of combination; in no case is any force destroyed.

Sometimes the separate tendencies of combined forces can only be theoretically distinguished: as when the motion of a projectile is a.n.a.lysed into a tendency to travel in the straight line of its discharge, and a tendency to fall straight to the ground. But sometimes a tendency can be isolated: as when,--after dropping a feather in some place sheltered from the wind, and watching it drift to and fro, as the air, offering unequal resistances to its uneven surface, counteracts its weight with varying success, until it slowly settles upon the ground,--we take it up and drop it again in a vacuum, when it falls like lead. Here we have the tendency of a certain cause (namely, the relation between the feather and the earth) free from counteraction: and this is called the _Elimination_ of the counteracting circ.u.mstances. In this case indeed there is physical elimination; whereas, in the case of a projectile, when we say that its actual motion is resolvable (neglecting the resistance of the air) into two tendencies, one in the line of discharge, the other earthwards, there is only theoretical elimination of either tendency, considered as counteracting the other; and this is more specifically called the _Resolution_ or a.n.a.lysis of the total effect into its component conditions. Now, Elimination and Resolution may be said to be the essential process of Induction in the widest sense of the term, as including the combination of Induction with Deduction.

The several conditions const.i.tuting any cause, then, by aiding or counteracting one another's tendencies, jointly determine the total effect. Hence, viewed in relation one to another, they may be said to stand in _Reciprocity_ or mutual influence. This relation at any moment is itself one of co-existence, though it is conceived with reference to a possible effect. As Kant says, all substances, as perceived in s.p.a.ce at the same time, are in reciprocal activity. And what is true of the world of things at any moment (as connected, say, by gravity), is true of any selected group of circ.u.mstances which we regard as the particular cause of any event to come. The use of the concept of reciprocity, then, lies in the a.n.a.lysis of a cause: we must not think of reciprocity as obtaining in the succession of cause and effect, as if the effect could turn back upon its cause; for as the effect arises its cause disappears, and is irrecoverable by Nature or Magic. There are many cases of rhythmic change and of moving equilibria, in which one movement or process produces another, and this produces something closely resembling the former, and so on in long series; as with the swing of a pendulum or the orbit of a planet: but these are series of cause and effect, not of reciprocity.

CHAPTER XV

INDUCTIVE METHOD

-- 1. It is necessary to describe briefly the process of investigating laws of causation, not with the notion of teaching any one the Art of Discovery, which each man pursues for himself according to his natural gifts and his experience in the methods of his own science, but merely to cast some light upon the contents of the next few chapters. Logic is here treated as a process of proof; proof supposes that some general proposition or hypothesis has been suggested as requiring proof; and the search for such propositions may spring from scientific curiosity or from practical interests.

We may, as Bain observes (_Logic_: B. iii. ch. 5), desire to detect a process of causation either (1) amidst circ.u.mstances that have no influence upon the process but only obscure it; as when, being pleased with a certain scent in a garden, we wish to know from what flower it rises; or, being attracted by the sound of some instrument in an orchestra, we desire to know which it is: or (2) amidst circ.u.mstances that alter the effect from what it would have been by the sole operation of some cause; as when the air deflects a falling feather; or in some more complex case, such as a rise or fall of prices that may extend over many years.

To begin with, we must form definite ideas as to what the phenomenon is that we are about to investigate; and in a case of any complexity this is best done by writing a detailed description of it: e.g., to investigate the cause of a recent fall of prices, we must describe exactly the course of the phenomenon, dating the period over which it extends, recording the successive fluctuations of prices, with their maxima and minima, and noting the cla.s.ses of goods or securities that were more or less affected, etc.

Then the first step of elimination (as Bain further observes) is "to a.n.a.lyse the situation mentally," in the light of a.n.a.logies suggested by our experience or previous knowledge. Dew, for example, is moisture formed upon the surface of bodies from no apparent source. But two possible sources are easily suggested by common experience: is it deposited from the air, like the moisture upon a mirror when we breathe upon it; or does it exude from the bodies themselves, like gum or turpentine? Or, again, as to a fall of prices, a little experience in business, or knowledge of Economics, readily suggests two possible explanations: either cheaper production in making goods or carrying them; or a scarcity of that in which the purchasing power of the chief commercial nations is directly expressed, namely, gold.

Having thus a.n.a.lysed the situation and considered the possibility of one, two, three, or more possible causes, we fix upon one of them for further investigation; that is to say, we frame an hypothesis that this is the cause. When an effect is given to find its cause, an inquirer nearly always begins his investigations by thus framing an hypothesis as to the cause.

The next step is to try to _verify_ this Hypothesis. This we may sometimes do by _varying the circ.u.mstances_ of the phenomenon, according to the Canons of direct Inductive Proof to be discussed in the next chapter; that is to say, by _observing_ or _experimenting_ in such a way as to get rid of or eliminate the obscuring or disturbing conditions.

Thus, to find out which flower in a garden gives a certain scent, it is usually enough to rely on observation, going up to the likely flowers one after the other and smelling them: at close quarters, the greater relative intensity of the scent is sufficiently decisive. Or we may resort to a sort of experiment, plucking a likely flower, as to which we frame the hypothesis (this is the cause), and carrying it to some place where the air is free from conflicting odours. Should observation or experiment disprove our first hypothesis we try a second; and so on until we succeed, or exhaust the known possibilities.

But if the phenomenon is so complex and extensive as a continuous fall of prices, direct observation or experiment is a useless or impossible method; and we must then resort to Deduction; that is, to indirect Induction. If, for example, we take the hypothesis that the fall is due to a scarcity of gold, we must show that there is a scarcity; what effect such a scarcity may be expected to have upon prices from the acknowledged laws of prices, and from the a.n.a.logy of other cases of an expanded or restricted currency; that this expectation agrees with the statistics of recent commerce: and finally, that the alternative hypothesis that the fall is due to cheaper production is not true; either because there has not been a sufficient cheapening of general production; or because, if there has been, the results to be rationally expected from it are not such as to agree with the statistics of recent commerce. (Ch. xviii.)

But now suppose that, a phenomenon having been suggested for explanation, we are unable at the time to think of any cause--to frame any hypothesis about it; we must then wait for the phenomenon to occur again, and, once more observing its course and accompaniments and trying to recall its antecedents, do our best to conceive an hypothesis, and proceed as before. Thus, in the first great epidemic of influenza, some doctors traced it to a deluge in China, others to a volcanic eruption near Java; some thought it a mild form of Asiatic plague, and others caught a specific microbe. As the disease often recurred, there were fresh opportunities of framing hypotheses; and the microbe was identified.

Again, the investigation may take a different form: given a supposed Cause to find its Effect; e.g., a new chemical element, to find what compounds it forms with other elements; or, the spots on the sun--have they any influence upon our weather?

Here, if the given cause be under control, as a new element may be, it is possible to try experiments with it according to the Canons of Inductive Proof. The inquirer may form some hypothesis or expectation as to the effects, to guide his observation of them, but will be careful not to hold his expectation so confidently as to falsify his observation of what actually happens.

But if the cause be, like the sun-spots, not under control, the inquirer will watch on all sides what events follow their appearance and development; he must watch for consequences of the new cause he is studying in many different circ.u.mstances, that his observations may satisfy the canons of proof. But he will also resort for guidance to deduction; arguing from the nature of the cause, if anything is known of its nature, what consequences may be expected, and comparing the results of this deduction with any consequent which he suspects to be connected with the cause. And if the results of deduction and observation agree, he will still consider whether the facts observed may not be due to some other cause.

A cause, however, may be under control and yet be too dangerous to experiment with; such as the effects of a poison--though, if too dangerous to experiment with upon man, it may be tried upon animals; or such as a proposed change of the const.i.tution by legislation; or even some minor Act of Parliament, for altering the Poor Law, or regulating the hours of labour. Here the first step must be deductive. We must ask what consequences are to be expected from the nature of the change (comparing it with similar changes), and from the laws of the special circ.u.mstances in which it is to operate? And sometimes we may partially verify our deduction by trying experiments upon a small scale or in a mild form. There are conflicting deductions as to the probable effect of giving Home Rule to Ireland; and experiments have been made in more or less similar cases, as in the Colonies and in some foreign countries. As to the proposal to make eight hours the legal limit of a day's labour in all trades, we have all tried to forecast the consequences of this; and by way of verification we might begin with nine hours; or we might induce some other country to try the experiment first. Still, no verification by experiments on a small scale, or in a mild form, or in somewhat similar yet different circ.u.mstances, can be considered logically conclusive. What proofs are conclusive we shall see in the following chapters.

-- 2. To begin with the conditions of direct Induction.--An Induction is an universal real proposition, based on observation, in reliance on the uniformity of Nature: when well ascertained, it is called a Law. Thus, that all life depends on the presence of oxygen is (1) an universal proposition; (2) a real one, since the 'presence of oxygen' is not connoted by 'life'; (3) it is based on observation; (4) it relies on the uniformity of Nature, since all cases of life have not been examined.

Such a proposition is here called 'an induction,' when it is inductively proved; that is, proved by facts, not merely deduced from more general premises (except the premise of Nature's uniformity): and by the 'process of induction' is meant the method of inductive proof. The phrase 'process of induction' is often used in another sense, namely for the inference or judgment by which such propositions are arrived at. But it is better to call this 'the process of hypothesis,' and to regard it as a preliminary to the process of induction (that is, proof), as furnis.h.i.+ng the hypothesis which, if it can stand the proper tests, becomes an induction or law.

-- 3. Inductive proofs are usually cla.s.sed as Perfect and Imperfect.

They are said to be perfect when all the instances within the scope of the given proposition have been severally examined, and the proposition has been found true in each case. But we have seen (chap. xiii. -- 2) that the instances included in universal propositions concerning Causes and Kinds cannot be exhaustively examined: we do not know all planets, all heat, all liquids, all life, etc.; and we never can, since a man's life is never long enough. It is only where the conditions of time, place, etc., are arbitrarily limited that examination can be exhaustive.

Perfect induction might show (say) that every member of the present House of Commons has two Christian names. Such an argument is sometimes exhibited as a Syllogism in Darapti with a Minor premise in U., which legitimates a Conclusion in A., thus:

A.B. to Z have two Christian names; A.B. to Z are all the present M.P.'s: ? All the present M.P.'s have two Christian names.

But in such an investigation there is no need of logical method to find the major premise; it is mere counting: and to carry out the syllogism is a hollow formality. Accordingly, our definition of Induction excludes the kind unfortunately called Perfect, by including in the notion of Induction a reliance on the uniformity of Nature; for this would be superfluous if every instance in question had been severally examined.

Imperfect Induction, then, is what we have to deal with: the method of showing the credibility of an universal real proposition by an examination of _some_ of the instances it includes, generally a small fraction of them.

-- 4. Imperfect Induction is either Methodical or Immethodical. Now, Method is procedure upon a principle; and if the method is to be precise and conclusive, the principle must be clear and definite.

There is a Geometrical Method, because the axioms of Geometry are clear and definite, and by their means, with the aid of definitions, laws are deduced of the equality of lines and angles and other relations of position and magnitude in s.p.a.ce. The process of proof is purely Deductive (the axioms and definitions being granted). Diagrams are used not as facts for observation, but merely to fix our attention in following the general argument; so that it matters little how badly they are drawn, as long as their divergence from the conditions of the proposition to be proved is not distracting. Even the appeal to "superposition" to prove the equality of magnitudes (as in Euclid I. 4), is not an appeal to observation, but to our judgment of what is implied in the foregoing conditions. Hence no inference is required from the special case to all similar ones; for they are all proved at once.

There is also, as we have seen, a method of Deductive Logic resting on the Principles of Consistency and the _Dictum de omni et nullo_. And we shall find that there is a method of Inductive Logic, resting on the principle of Causation.

But there are a good many general propositions, more or less trustworthy within a certain range of conditions, which cannot be methodically proved for want of a precise principle by which they may be tested; and they, therefore, depend upon Immethodical Induction, that is, upon the examination of as many instances as can be found, relying for the rest upon the undefinable principle of the Uniformity of Nature, since we are not able to connect them with any of its definite modes enumerated in chap. xiii. -- 7. To this subject we shall return in chap. xix., after treating of Methodical Induction, or the means of determining that a relation of events is of the nature of cause and effect, because the relation can be shown to have the marks of causation, or some of them.