CHAPTER XI.

PHILOSOPHY OF INDUCTIVE INFERENCE.

We have inquired into the nature of perfect induction, whereby we pass backwards from certain observed combinations of events, to the logical conditions governing such combinations. We have also investigated the grounds of that theory of probability, which must be our guide when we leave certainty behind, and dilute knowledge with ignorance. There is now before us the difficult task of endeavouring to decide how, by the aid of that theory, we can ascend from the facts to the laws of nature; and may then with more or less success anticipate the future course of events. All our knowledge of natural objects must be ultimately derived from observation, and the difficult question arises--How can we ever know anything which we have not directly observed through one of our senses, the apertures of the mind? The utility of reasoning is to assure ourselves that, at a determinate time and place, or under specified conditions, a certain phenomenon will be observed. When we can use our senses and perceive that the phenomenon does occur, reasoning is superfluous. If the senses cannot be used, because the event is in the future, or out of reach, how can reasoning take their place? Apparently, at least, we must infer the unknown from the known, and the mind must itself create an addition to the sum of knowledge. But I hold that it is quite impossible to make any real additions to the contents of our knowledge, except through new impressions upon the senses, or upon some seat of feeling. I shall attempt to show that inference, whether inductive or deductive, is never more than an unfolding of the contents of our experience, and that it always proceeds upon the assumption that the future and the unperceived will be governed by the same conditions as the past and the perceived, an assumption which will often prove to be mistaken.

In inductive as in deductive reasoning the conclusion never passes beyond the premises. Reasoning adds no more to the implicit contents of our knowledge, than the arrangement of the specimens in a museum adds to the number of those specimens. Arrangement adds to our knowledge in a certain sense: it allows us to perceive the similarities and peculiarities of the specimens, and on the assumption that the museum is an adequate representation of nature, it enables us to judge of the prevailing forms of natural objects. Bacon’s first aphorism holds perfectly true, that man knows nothing but what he has observed, provided that we include his whole sources of experience, and the whole implicit contents of his knowledge. Inference but unfolds the hidden meaning of our observations, and *the theory of probability shows how far we go beyond our data in assuming that new specimens will resemble the old ones*, or that the future may be regarded as proceeding uniformly with the past.

*Various Classes of Inductive Truths.*

It will be desirable, in the first place, to distinguish between the several kinds of truths which we endeavour to establish by induction. Although there is a certain common and universal element in all our processes of reasoning, yet diversity arises in their application. Similarity of condition between the events from which we argue, and those to which we argue, must always be the ground of inference; but this similarity may have regard either to time or place, or the simple logical combination of events, or to any conceivable junction of circumstances involving quality, time, and place. Having met with many pieces of substance possessing ductility and a bright yellow colour, and having discovered, by perfect induction, that they all possess a high specific gravity, and a freedom from the corrosive action of acids, we are led to expect that every piece of substance, possessing like ductility and a similar yellow colour, will have an equally high specific gravity, and a like freedom from corrosion by acids. This is a case of the coexistence of qualities; for the character of the specimens examined alters not with time nor place.

In a second class of cases, time will enter as a principal ground of similarity. When we hear a clock pendulum beat time after time, at equal intervals, and with a uniform sound, we confidently expect that the stroke will continue to be repeated uniformly. A comet having appeared several times at nearly equal intervals, we infer that it will probably appear again at the end of another like interval. A man who has returned home evening after evening for many years, and found his house standing, may, on like grounds, expect that it will be standing the next evening, and on many succeeding evenings. Even the continuous existence of an object in an unaltered state, or the finding again of that which we have hidden, is but a matter of inference depending on experience.

A still larger and more complex class of cases involves the relations of space, in addition to those of time and quality. Having observed that every triangle drawn upon the diameter of a circle, with its apex upon the circumference, apparently contains a right angle, we may ascertain that all triangles in similar circumstances will contain right angles. This is a case of pure space reasoning, apart from circumstances of time or quality, and it seems to be governed by different principles of reasoning. I shall endeavour to show, however, that geometrical reasoning differs but in degree from that which applies to other natural relations.

*The Relation of Cause and Effect.*

In a very large part of the scientific investigations which must be considered, we deal with events which follow from previous events, or with existences which succeed existences. Science, indeed, might arise even were material nature a fixed and changeless whole. Endow mind with the power to travel about, and compare part with part, and it could certainly draw inferences concerning the similarity of forms, the coexistence of qualities, or the preponderance of a particular kind of matter in a changeless world. A solid universe, in at least approximate equilibrium, is not inconceivable, and then the relation of cause and effect would evidently be no more than the relation of before and after. As nature exists, however, it is a progressive existence, ever moving and changing as time, the great independent variable, proceeds. Hence it arises that we must continually compare what is happening now with what happened a moment before, and a moment before that moment, and so on, until we reach indefinite periods of past time. A comet is seen moving in the sky, or its constituent particles illumine the heavens with their tails of fire. We cannot explain the present movements of such a body without supposing its prior existence, with a definite amount of energy and a definite direction of motion; nor can we validly suppose that our task is concluded when we find that it came wandering to our solar system through the unmeasured vastness of surrounding space. Every event must have a cause, and that cause again a cause, until we are lost in the obscurity of the past, and are driven to the belief in one First Cause, by whom the course of nature was determined.

*Fallacious Use of the Term Cause.*

The words Cause and Causation have given rise to infinite trouble and obscurity, and have in no slight degree retarded the progress of science. From the time of Aristotle, the work of philosophy has been described as the discovery of the causes of things, and Francis Bacon adopted the notion when he said “*vere scire esse per causas scire*.” Even now it is not uncommonly supposed that the knowledge of causes is something different from other knowledge, and consists, as it were, in getting possession of the keys of nature. A single word may thus act as a spell, and throw the clearest intellect into confusion, as I have often thought that Locke was thrown into confusion when endeavouring to find a meaning for the word *power*.[128] In Mill’s *System of Logic* the term *cause* seems to have re-asserted its old noxious power. Not only does Mill treat the Laws of Causation as almost coextensive with science, but he so uses the expression as to imply that when once we pass within the circle of causation we deal with certainties.

[128] *Essay concerning Human Understanding*, bk. ii. chap. xxi.

The philosophical danger which attaches to the use of this word may be thus described. A cause is defined as the necessary or invariable antecedent of an event, so that when the cause exists the effect will also exist or soon follow. If then we know the cause of an event, we know what will certainly happen; and as it is implied that science, by a proper experimental method, may attain to a knowledge of causes, it follows that experience may give us a certain knowledge of future events. But nothing is more unquestionable than that finite experience can never give us certain knowledge of the future, so that either a cause is not an invariable antecedent, or else we can never gain certain knowledge of causes. The first horn of this dilemma is hardly to be accepted. Doubtless there is in nature some invariably acting mechanism, such that from certain fixed conditions an invariable result always emerges. But we, with our finite minds and short experience, can never penetrate the mystery of those existences which embody the Will of the Creator, and evolve it throughout time. We are in the position of spectators who witness the productions of a complicated machine, but are not allowed to examine its intimate structure. We learn what does happen and what does appear, but if we ask for the reason, the answer would involve an infinite depth of mystery. The simplest bit of matter, or the most trivial incident, such as the stroke of two billiard balls, offers infinitely more to learn than ever the human intellect can fathom. The word cause covers just as much untold meaning as any of the words *substance*, *matter*, *thought*, *existence*.

*Confusion of Two Questions.*

The subject is much complicated, too, by the confusion of two distinct questions. An event having happened, we may ask--

(1) Is there any cause for the event? (2) Of what kind is that cause?

No one would assert that the mind possesses any faculty capable of inferring, prior to experience, that the occurrence of a sudden noise with flame and smoke indicates the combustion of a black powder, formed by the mixture of black, white, and yellow powders. The greatest upholder of *à priori* doctrines will allow that the particular aspect, shape, size, colour, texture, and other qualities of a cause must be gathered through the senses.

The question whether there is any cause at all for an event, is of a totally different kind. If an explosion could happen without any prior existing conditions, it must be a new creation--a distinct addition to the universe. It may be plausibly held that we can imagine neither the creation nor annihilation of anything. As regards matter, this has long been held true; as regards force, it is now almost universally assumed as an axiom that energy can neither come into nor go out of existence without distinct acts of Creative Will. That there exists any instinctive belief to this effect, indeed, seems doubtful. We find Lucretius, a philosopher of the utmost intellectual power and cultivation, gravely assuming that his raining atoms could turn aside from their straight paths in a self-determining manner, and by this spontaneous origination of energy determine the form of the universe.[129] Sir George Airy, too, seriously discussed the mathematical conditions under which a perpetual motion, that is, a perpetual source of self-created energy, might exist.[130] The larger part of the philosophic world has long held that in mental acts there is free will--in short, self-causation. It is in vain to attempt to reconcile this doctrine with that of an intuitive belief in causation, as Sir W. Hamilton candidly allowed.

[129] *De Rerum Natura*, bk. ii. ll. 216–293.

[130] *Cambridge Philosophical Transactions* (1830), vol. iii. pp. 369–372.

It is obvious, moreover, that to assert the existence of a cause for every event cannot do more than remove into the indefinite past the inconceivable fact and mystery of creation. At any given moment matter and energy were equal to what they are at present, or they were not; if equal, we may make the same inquiry concerning any other moment, however long prior, and we are thus obliged to accept one horn of the dilemma--existence from infinity, or creation at some moment. This is but one of the many cases in which we are compelled to believe in one or other of two alternatives, both inconceivable. My present purpose, however, is to point out that we must not confuse this supremely difficult question with that into which inductive science inquires on the foundation of facts. By induction we gain no certain knowledge; but by observation, and the inverse use of deductive reasoning, we estimate the probability that an event which has occurred was preceded by conditions of specified character, or that such conditions will be followed by the event.

*Definition of the Term Cause.*

Clear definitions of the word cause have been given by several philosophers. Hobbes has said, “A cause is the sum or aggregate of all such accidents, both in the agents and the patients, as concur in the producing of the effect propounded; all which existing together, it cannot be understood but that the effect existeth with them; or that it can possibly exist if any of them be absent.” Brown, in his *Essay on Causation*, gave a nearly corresponding statement. “A cause,” he says,[131] “may be defined to be the object or event which immediately precedes any change, and which existing again in similar circumstances will be always immediately followed by a similar change.” Of the kindred word *power*, he likewise says:[132] “Power is nothing more than that invariableness of antecedence which is implied in the belief of causation.”

[131] *Observations on the Nature and Tendency of the Doctrine of Mr. Hume, concerning the Relation of Cause and Effect.* Second ed. p. 44.

[132] Ibid. p. 97.

These definitions may be accepted with the qualification that our knowledge of causes in such a sense can be probable only. The work of science consists in ascertaining the combinations in which phenomena present themselves. Concerning every event we shall have to determine its probable conditions, or the group of antecedents from which it probably follows. An antecedent is anything which exists prior to an event; a consequent is anything which exists subsequently to an antecedent. It will not usually happen that there is any probable connection between an antecedent and consequent. Thus nitrogen is an antecedent to the lighting of a common fire; but it is so far from being a cause of the lighting, that it renders the combustion less active. Daylight is an antecedent to all fires lighted during the day, but it probably has no appreciable effect upon their burning. But in the case of any given event it is usually possible to discover a certain number of antecedents which seem to be always present, and with more or less probability we conclude that when they exist the event will follow.

Let it be observed that the utmost latitude is at present enjoyed in the use of the term *cause*. Not only may a cause be an existent thing endowed with powers, as oxygen is the cause of combustion, gunpowder the cause of explosion, but the very absence or removal of a thing may also be a cause. It is quite correct to speak of the dryness of the Egyptian atmosphere, or the absence of moisture, as being the cause of the preservation of mummies, and other remains of antiquity. The cause of a mountain elevation, Ingleborough for instance, is the excavation of the surrounding valleys by denudation. It is not so usual to speak of the existence of a thing at one moment as the cause of its existence at the next, but to me it seems the commonest case of causation which can occur. The cause of motion of a billiard ball may be the stroke of another ball; and recent philosophy leads us to look upon all motions and changes, as but so many manifestations of prior existing energy. In all probability there is no creation of energy and no destruction, so that as regards both mechanical and molecular changes, the cause is really the manifestation of existing energy. In the same way I see not why the prior existence of matter is not also a cause as regards its subsequent existence. All science tends to show us that the existence of the universe in a particular state at one moment, is the condition of its existence at the next moment, in an apparently different state. When we analyse the meaning which we can attribute to the word *cause*, it amounts to the existence of suitable portions of matter endowed with suitable quantities of energy. If we may accept Horne Tooke’s assertion, *cause* has etymologically the meaning of *thing before*. Though, indeed, the origin of the word is very obscure, its derivatives, the Italian *cosa*, and the French *chose*, mean simply *thing*. In the German equivalent *ursache*, we have plainly the original meaning of *thing before*, the *sache* denoting “interesting or important object,” the English *sake*, and *ur* being the equivalent of the English *ere*, *before*. We abandon, then, both etymology and philosophy, when we attribute to the *laws of causation* any meaning beyond that of the *conditions* under which an event may be expected to happen, according to our observation of the previous course of nature.

I have no objection to use the words cause and causation, provided they are never allowed to lead us to imagine that our knowledge of nature can attain to certainty. I repeat that if a cause is an invariable and necessary condition of an event, we can never know certainly whether the cause exists or not. To us, then, a cause is not to be distinguished from the group of positive or negative conditions which, with more or less probability, precede an event. In this sense, there is no particular difference between knowledge of causes and our general knowledge of the succession of combinations, in which the phenomena of nature are presented to us, or found to occur in experimental inquiry.

*Distinction of Inductive and Deductive Results.*

We must carefully avoid confusing together inductive investigations which terminate in the establishment of general laws, and those which seem to lead directly to the knowledge of future particular events. That process only can be called induction which gives general laws, and it is by the subsequent employment of deduction that we anticipate particular events. If the observation of a number of cases shows that alloys of metals fuse at lower temperatures than their constituent metals, I may with more or less probability draw a general inference to that effect, and may thence deductively ascertain the probability that the next alloy examined will fuse at a lower temperature than its constituents. It has been asserted, indeed, by Mill,[133] and partially admitted by Mr. Fowler,[134] that we can argue directly from case to case, so that what is true of some alloys will be true of the next. Professor Bain has adopted the same view of reasoning. He thinks that Mill has extricated us from the dead lock of the syllogism and effected a total revolution in logic. He holds that reasoning from particulars to particulars is not only the usual, the most obvious and the most ready method, but that it is the type of reasoning which best discloses the real process.[135] Doubtless, this is the usual result of our reasoning, regard being had to degrees of probability; but these logicians fail entirely to give any explanation of the process by which we get from case to case.

[133] *System of Logic*, bk. II. chap, iii.

[134] *Inductive Logic*, pp. 13, 14.

[135] Bain, *Deductive Logic*, pp. 208, 209.

It may be allowed that the knowledge of future particular events is the main purpose of our investigations, and if there were any process of thought by which we could pass directly from event to event without ascending into general truths, this method would be sufficient, and certainly the briefest. It is true, also, that the laws of mental association lead the mind always to expect the like again in apparently like circumstances, and even animals of very low intelligence must have some trace of such powers of association, serving to guide them more or less correctly, in the absence of true reasoning faculties. But it is the purpose of logic, according to Mill, to ascertain whether inferences have been correctly drawn, rather than to discover them.[136] Even if we can, then, by habit, association, or any rude process of inference, infer the future directly from the past, it is the work of logic to analyse the conditions on which the correctness of this inference depends. Even Mill would admit that such analysis involves the consideration of general truths,[137] and in this, as in several other important points, we might controvert Mill’s own views by his own statements. It seems to me undesirable in a systematic work like this to enter into controversy at any length, or to attempt to refute the views of other logicians. But I shall feel bound to state, in a separate publication, my very deliberate opinion that many of Mill’s innovations in logical science, and especially his doctrine of reasoning from particulars to particulars, are entirely groundless and false.

[136] *System of Logic.* Introduction, § 4. Fifth ed. pp. 8, 9.

[137] Ibid. bk. II. chap. iii. § 5, pp. 225, &c.

*The Grounds of Inductive Inference.*

I hold that in all cases of inductive inference we must invent hypotheses, until we fall upon some hypothesis which yields deductive results in accordance with experience. Such accordance renders the chosen hypothesis more or less probable, and we may then deduce, with some degree of likelihood, the nature of our future experience, on the assumption that no arbitrary change takes place in the conditions of nature. We can only argue from the past to the future, on the general principle set forth in this work, that what is true of a thing will be true of the like. So far then as one object or event differs from another, all inference is impossible, particulars as particulars can no more make an inference than grains of sand can make a rope. We must always rise to something which is general or same in the cases, and assuming that sameness to be extended to new cases we learn their nature. Hearing a clock tick five thousand times without exception or variation, we adopt the very probable hypothesis that there is some invariably acting machine which produces those uniform sounds, and which will, in the absence of change, go on producing them. Meeting twenty times with a bright yellow ductile substance, and finding it always to be very heavy and incorrodible, I infer that there was some natural condition which tended in the creation of things to associate these properties together, and I expect to find them associated in the next instance. But there always is the possibility that some unknown change may take place between past and future cases. The clock may run down, or be subject to a hundred accidents altering its condition. There is no reason in the nature of things, so far as known to us, why yellow colour, ductility, high specific gravity, and incorrodibility, should always be associated together, and in other cases, if not in this, men’s expectations have been deceived. Our inferences, therefore, always retain more or less of a hypothetical character, and are so far open to doubt. Only in proportion as our induction approximates to the character of perfect induction, does it approximate to certainty. The amount of uncertainty corresponds to the probability that other objects than those examined may exist and falsity our inferences; the amount of probability corresponds to the amount of information yielded by our examination; and the theory of probability will be needed to prevent us from over-estimating or under-estimating the knowledge we possess.

*Illustrations of the Inductive Process.*

To illustrate the passage from the known to the apparently unknown, let us suppose that the phenomena under investigation consist of numbers, and that the following six numbers being exhibited to us, we are required to infer the character of the next in the series:--

5, 15, 35, 45, 65, 95.

The question first of all arises, How may we describe this series of numbers? What is uniformly true of them? The reader cannot fail to perceive at the first glance that they all end in five, and the problem is, from the properties of these six numbers, to infer the properties of the next number ending in five. If we test their properties by the process of perfect induction, we soon perceive that they have another common property, namely that of being *divisible by five without remainder*. May we then assert that the next number ending in five is also divisible by five, and, if so, upon what grounds? Or extending the question, Is every number ending in five divisible by five? Does it follow that because six numbers obey a supposed law, therefore 376,685,975 or any other number, however large, obeys the law? I answer *certainly not*. The law in question is undoubtedly true; but its truth is not proved by any finite number of examples. All that these six numbers can do is to suggest to my mind the possible existence of such a law; and I then ascertain its truth, by proving deductively from the rules of decimal numeration, that any number ending in five must be made up of multiples of five, and must therefore be itself a multiple.

To make this more plain, let the reader now examine the numbers--

7, 17, 37, 47, 67, 97.

They all end in 7 instead of 5, and though not at equal intervals, the intervals are the same as in the previous case. After consideration, the reader will perceive that these numbers all agree in being *prime numbers*, or multiples of unity only. May we then infer that the next, or any other number ending in 7, is a prime number? Clearly not, for on trial we find that 27, 57, 117 are not primes. Six instances, then, treated empirically, lead us to a true and universal law in one case, and mislead us in another case. We ought, in fact, to have no confidence in any law until we have treated it deductively, and have shown that from the conditions supposed the results expected must ensue. No one can show from the principles of number, that numbers ending in 7 should be primes.

From the history of the theory of numbers some good examples of false induction can be adduced. Taking the following series of prime numbers,

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c.,

it will be found that they all agree in being values of the general expression *x*^{2} + *x* + 41, putting for *x* in succession the values, 0, 1, 2, 3, 4, &c. We seem always to obtain a prime number, and the induction is apparently strong, to the effect that this expression always will give primes. Yet a few more trials disprove this false conclusion. Put *x* = 40, and we obtain 40 × 40 + 40 + 41, or 41 × 41. Such a failure could never have happened, had we shown any deductive reason why *x*^{2} + *x* + 41 should give primes.

There can be no doubt that what here happens with forty instances, might happen with forty thousand or forty million instances. An apparent law never once failing up to a certain point may then suddenly break down, so that inductive reasoning, as it has been described by some writers, can give no sure knowledge of what is to come. Babbage pointed out, in his Ninth Bridgewater Treatise, that a machine could be constructed to give a perfectly regular series of numbers through a vast series of steps, and yet to break the law of progression suddenly at any required point. No number of particular cases as particulars enables us to pass by inference to any new case. It is hardly needful to inquire here what can be inferred from an infinite series of facts, because they are never practically within our power; but we may unhesitatingly accept the conclusion, that no finite number of instances can ever prove a general law, or can give us certain knowledge of even one other instance.

General mathematical theorems have indeed been discovered by the observation of particular cases, and may again be so discovered. We have Newton’s own statement, to the effect that he was thus led to the all-important Binomial Theorem, the basis of the whole structure of mathematical analysis. Speaking of a certain series of terms, expressing the area of a circle or hyperbola, he says: “I reflected that the denominators were in arithmetical progression; so that only the numerical co-efficients of the numerators remained to be investigated. But these, in the alternate areas, were the figures of the powers of the number eleven, namely 11^{0}, 11^{1}, 11^{2}, 11^{3}, 11^{4}; that is, in the first 1; in the second 1, 1; in the third 1, 2, 1; in the fourth 1, 3, 3, 1; in the fifth 1, 4, 6, 4, 1.[138] I inquired, therefore, in what manner all the remaining figures could be found from the first two; and I found that if the first figure be called *m*, all the rest could be found by the continual multiplication of the terms of the formula

((*m* - 0)/1) × ((*m* - 1)/2) × ((*m* - 2)/3) × ((*m* - 3)/4) × &c.”[139]

[138] These are the figurate numbers considered in pages 183, 187, &c.

[139] *Commercium Epistolicum.* *Epistola ad Oldenburgum*, Oct. 24, 1676. Horsley’s *Works of Newton*, vol. iv. p. 541. See De Morgan in *Penny Cyclopædia*, art. “Binomial Theorem,” p. 412.

It is pretty evident, from this most interesting statement, that Newton, having simply observed the succession of the numbers, tried various formulæ until he found one which agreed with them all. He was so little satisfied with this process, however, that he verified particular results of his new theorem by comparison with the results of common multiplication, and the rule for the extraction of the square root. Newton, in fact, gave no demonstration of his theorem; and the greatest mathematicians of the last century, James Bernoulli, Maclaurin, Landen, Euler, Lagrange, &c., occupied themselves with discovering a conclusive method of deductive proof.

There can be no doubt that in geometry also discoveries have been suggested by direct observation. Many of the now trivial propositions of Euclid’s Elements were probably thus discovered, by the ancient Greek geometers; and we have pretty clear evidence of this in the Commentaries of Proclus.[140] Galileo was the first to examine the remarkable properties of the cycloid, the curve described by a point in the circumference of a wheel rolling on a plane. By direct observation he ascertained that the area of the curve is apparently three times that of the generating circle or wheel, but he was unable to prove this exactly, or to verify it by strict geometrical reasoning. Sir George Airy has recorded a curious case, in which he fell accidentally by trial on a new geometrical property of the sphere.[141] But discovery in such cases means nothing more than suggestion, and it is always by pure deduction that the general law is really established. As Proclus puts it, *we must pass from sense to consideration*.

[140] Bk. ii. chap. iv.

[141] *Philosophical Transactions* (1866), vol. 146, p. 334.

[Illustration]

Given, for instance, the series of figures in the accompanying diagram, measurement will show that the curved lines approximate to semicircles, and the rectilinear figures to right-angled triangles. These figures may seem to suggest to the mind the general law that angles inscribed in semicircles are right angles; but no number of instances, and no possible accuracy of measurement would really establish the truth of that general law. Availing ourselves of the suggestion furnished by the figures, we can only investigate deductively the consequences which flow from the definition of a circle, until we discover among them the property of containing right angles. Persons have thought that they had discovered a method of trisecting angles by plane geometrical construction, because a certain complex arrangement of lines and circles had appeared to trisect an angle in every case tried by them, and they inferred, by a supposed act of induction, that it would succeed in all other cases. De Morgan has recorded a proposed mode of trisecting the angle which could not be discriminated by the senses from a true general solution, except when it was applied to very obtuse angles.[142] In all such cases, it has always turned out either that the angle was not trisected at all, or that only certain particular angles could be thus trisected. The trisectors were misled by some apparent or special coincidence, and only deductive proof could establish the truth and generality of the result. In this particular case, deductive proof shows that the problem attempted is impossible, and that angles generally cannot be trisected by common geometrical methods.

[142] *Budget of Paradoxes*, p. 257.

*Geometrical Reasoning.*

This view of the matter is strongly supported by the further consideration of geometrical reasoning. No skill and care could ever enable us to verify absolutely any one geometrical proposition. Rousseau, in his *Emile*, tells us that we should teach a child geometry by causing him to measure and compare figures by superposition. While a child was yet incapable of general reasoning, this would doubtless be an instructive exercise; but it never could teach geometry, nor prove the truth of any one proposition. All our figures are rude approximations, and they may happen to seem unequal when they should be equal, and equal when they should be unequal. Moreover figures may from chance be equal in case after case, and yet there may be no general reason why they should be so. The results of deductive geometrical reasoning are absolutely certain, and are either exactly true or capable of being carried to any required degree of approximation. In a perfect triangle, the angles must be equal to one half-revolution precisely; even an infinitesimal divergence would be impossible; and I believe with equal confidence, that however many are the angles of a figure, provided there are no re-entrant angles, the sum of the angles will be precisely and absolutely equal to twice as many right-angles as the figure has sides, less by four right-angles. In such cases, the deductive proof is absolute and complete; empirical verification can at the most guard against accidental oversights.

There is a second class of geometrical truths which can only be proved by approximation; but, as the mind sees no reason why that approximation should not always go on, we arrive at complete conviction. We thus learn that the surface of a sphere is equal exactly to two-thirds of the whole surface of the circumscribing cylinder, or to four times the area of the generating circle. The area of a parabola is exactly two-thirds of that of the circumscribing parallelogram. The area of the cycloid is exactly three times that of the generating circle. These are truths that we could never ascertain, nor even verify by observation; for any finite amount of difference, less than what the senses can discern, would falsify them.

There are geometrical relations again which we cannot assign exactly, but can carry to any desirable degree of approximation. The ratio of the circumference to the diameter of a circle is that of 3·14159265358979323846.... to 1, and the approximation may be carried to any extent by the expenditure of sufficient labour. Mr. W. Shanks has given the value of this natural constant, known as π, to the extent of 707 places of decimals.[143] Some years since, I amused myself by trying how near I could get to this ratio, by the careful use of compasses, and I did not come nearer than 1 part in 540. We might imagine measurements so accurately executed as to give us eight or ten places correctly. But the power of the hands and senses must soon stop, whereas the mental powers of deductive reasoning can proceed to an unlimited degree of approximation. Geometrical truths, then, are incapable of verification; and, if so, they cannot even be learnt by observation. How can I have learnt by observation a proposition of which I cannot even prove the truth by observation, when I am in possession of it? All that observation or empirical trial can do is to suggest propositions, of which the truth may afterwards be proved deductively.

[143] *Proceedings of the Royal Society* (1872–3), vol. xxi. p. 319.

If Viviani’s story is to be believed, Galileo endeavoured to satisfy himself about the area of the cycloid by cutting out several large cycloids in pasteboard, and then comparing the areas of the curve and the generating circle by weighing them. In every trial the curve seemed to be rather less than three times the circle, so that Galileo, we are told, began to suspect that the ratio was not precisely 3 to 1. It is quite clear, however, that no process of weighing or measuring could ever prove truths like these, and it remained for Torricelli to show what his master Galileo had only guessed at.[144]

[144] *Life of Galileo*, Society for the Diffusion of Useful Knowledge, p. 102.

Much has been said about the peculiar certainty of mathematical reasoning, but it is only certainty of deductive reasoning, and equal certainty attaches to all correct logical deduction. If a triangle be right-angled, the square on the hypothenuse will undoubtedly equal the sum of the two squares on the other sides; but I can never be sure that a triangle is right-angled: so I can be certain that nitric acid will not dissolve gold, provided I know that the substances employed really correspond to those on which I tried the experiment previously. Here is like certainty of inference, and like doubt as to the facts.

*Discrimination of Certainty and Probability.*

We can never recur too often to the truth that our knowledge of the laws and future events of the external world is only probable. The mind itself is quite capable of possessing certain knowledge, and it is well to discriminate carefully between what we can and cannot know with certainty. In the first place, whatever feeling is actually present to the mind is certainly known to that mind. If I see blue sky, I may be quite sure that I do experience the sensation of blueness. Whatever I do feel, I do feel beyond all doubt. We are indeed very likely to confuse what we really feel with what we are inclined to associate with it, or infer inductively from it; but the whole of our consciousness, as far as it is the result of pure intuition and free from inference, is certain knowledge beyond all doubt.

In the second place, we may have certainty of inference; the fundamental laws of thought, and the rule of substitution (p. 9), are certainly true; and if my senses could inform me that A was indistinguishable in colour from B, and B from C, then I should be equally certain that A was indistinguishable from C. In short, whatever truth there is in the premises, I can certainly embody in their correct logical result. But the certainty generally assumes a hypothetical character. I never can be quite sure that two colours are exactly alike, that two magnitudes are exactly equal, or that two bodies whatsoever are identical even in their apparent qualities. Almost all our judgments involve quantitative relations, and, as will be shown in succeeding chapters, we can never attain exactness and certainty where continuous quantity enters. Judgments concerning discontinuous quantity or numbers, however, allow of certainty; I may establish beyond doubt, for instance, that the difference of the squares of 17 and 13 is the product of 17 + 13 and 17 - 13, and is therefore 30 × 4, or 120.

Inferences which we draw concerning natural objects are never certain except in a hypothetical point of view. It might seem to be certain that iron is magnetic, or that gold is incapable of solution in nitric acid; but, if we carefully investigate the meanings of these statements, they will be found to involve no certainty but that of consciousness and that of hypothetical inference. For what do I mean by iron or gold? If I choose a remarkable piece of yellow substance, call it gold, and then immerse it in a liquid which I call nitric acid, and find that there is no change called solution, then consciousness has certainly informed me that, with my meaning of the terms, “Gold is insoluble in nitric acid.” I may further be certain of something else; for if this gold and nitric acid remain what they were, I may be sure there will be no solution on again trying the experiment. If I take other portions of gold and nitric acid, and am sure that they really are identical in properties with the former portions, I can be certain that there will be no solution. But at this point my knowledge becomes purely hypothetical; for how can I be sure without trial that the gold and acid are really identical in nature with what I formerly called gold and nitric acid. How do I know gold when I see it? If I judge by the apparent qualities--colour, ductility, specific gravity, &c., I may be misled, because there may always exist a substance which to the colour, ductility, specific gravity, and other specified qualities, joins others which we do not expect. Similarly, if iron is magnetic, as shown by an experiment with objects answering to those names, then all iron is magnetic, meaning all pieces of matter identical with my assumed piece. But in trying to identify iron, I am always open to mistake. Nor is this liability to mistake a matter of speculation only.[145]

[145] Professor Bowen has excellently stated this view. *Treatise on Logic.* Cambridge, U.S.A., 1866, p. 354.

The history of chemistry shows that the most confident inferences may have been falsified by the confusion of one substance with another. Thus strontia was never discriminated from baryta until Klaproth and Haüy detected differences between some of their properties. Accordingly chemists must often have inferred concerning strontia what was only true of baryta, and *vice versâ*. There is now no doubt that the recently discovered substances, cæsium and rubidium, were long mistaken for potassium.[146] Other elements have often been confused together--for instance, tantalum and niobium; sulphur and selenium; cerium, lanthanum, and didymium; yttrium and erbium.

[146] Roscoe’s *Spectrum Analysis*, 1st edit., p. 98.

Even the best known laws of physical science do not exclude false inference. No law of nature has been better established than that of universal gravitation, and we believe with the utmost confidence that any body capable of affecting the senses will attract other bodies, and fall to the earth if not prevented. Euler remarks that, although he had never made trial of the stones which compose the church of Magdeburg, yet he had not the least doubt that all of them were heavy, and would fall if unsupported. But he adds, that it would be extremely difficult to give any satisfactory explanation of this confident belief.[147] The fact is, that the belief ought not to amount to certainty until the experiment has been tried, and in the meantime a slight amount of uncertainty enters, because we cannot be sure that the stones of the Magdeburg Church resemble other stones in all their properties.

[147] Euler’s *Letters to a German Princess*, translated by Hunter. 2nd ed., vol. ii. pp. 17, 18.

In like manner, not one of the inductive truths which men have established, or think they have established, is really safe from exception or reversal. Lavoisier, when laying the foundations of chemistry, met with so many instances tending to show the existence of oxygen in all acids, that he adopted a general conclusion to that effect, and devised the name oxygen accordingly. He entertained no appreciable doubt that the acid existing in sea salt also contained oxygen;[148] yet subsequent experience falsified his expectations. This instance refers to a science in its infancy, speaking relatively to the possible achievements of men. But all sciences are and ever will remain in their infancy, relatively to the extent and complexity of the universe which they undertake to investigate. Euler expresses no more than the truth when he says that it would be impossible to fix on any one thing really existing, of which we could have so perfect a knowledge as to put us beyond the reach of mistake.[149] We may be quite certain that a comet will go on moving in a similar path *if* all circumstances remain the same as before; but if we leave out this extensive qualification, our predictions will always be subject to the chance of falsification by some unexpected event, such as the division of Biela’s comet or the interference of an unknown gravitating body.

[148] Lavoisier’s *Chemistry*, translated by Kerr. 3rd ed., pp. 114, 121, 123.

[149] Euler’s *Letters*, vol. ii. p. 21.

Inductive inference might attain to certainty if our knowledge of the agents existing throughout the universe were complete, and if we were at the same time certain that the same Power which created the universe would allow it to proceed without arbitrary change. There is always a possibility of causes being in existence without our knowledge, and these may at any moment produce an unexpected effect. Even when by the theory of probabilities we succeed in forming some notion of the comparative confidence with which we should receive inductive results, it yet appears to me that we must make an assumption. Events come out like balls from the vast ballot-box of nature, and close observation will enable us to form some notion, as we shall see in the next chapter, of the contents of that ballot-box. But we must still assume that, between the time of an observation and that to which our inferences relate, no change in the ballot-box has been made.