CHAPTER XXXI.

REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.

Before concluding a work on the Principles of Science, it will not be inappropriate to add some remarks upon the limits and ultimate bearings of the knowledge which we may acquire by the employment of scientific method. All science consists, it has several times been stated, in the detection of identities in the action of natural agents. The purpose of inductive inquiry is to ascertain the apparent existence of necessary connection between causes and effects, expressed in the form of natural laws. Now so far as we thus learn the invariable course of nature, the future becomes the necessary sequel of the present, and we are brought beneath the sway of powers with which nothing can interfere.

By degrees it is found, too, that the chemistry of organised substances is not entirely separated from, but is continuous with, that of earth and stones. Life seems to be nothing but a special form of energy which is manifested in heat and electricity and mechanical force. The time may come, it almost seems, when the tender mechanism of the brain will be traced out, and every thought reduced to the expenditure of a determinate weight of nitrogen and phosphorus. No apparent limit exists to the success of scientific method in weighing and measuring, and reducing beneath the sway of law, the phenomena both of matter and of mind. And if mental phenomena be thus capable of treatment by the balance and the micrometer, can we any longer hold that mind is distinct from matter? Must not the same inexorable reign of law which is apparent in the motions of brute matter be extended to the subtle feelings of the human heart? Are not plants and animals, and ultimately man himself, merely crystals, as it were, of a complicated form? If so, our boasted free will becomes a delusion, moral responsibility a fiction, spirit a mere name for the more curious manifestations of material energy. All that happens, whether right or wrong, pleasurable or painful, is but the outcome of the necessary relations of time and space and force.

Materialism seems, then, to be the coming religion, and resignation to the nonentity of human will the only duty. Such may not generally be the reflections of men of science, but I believe that we may thus describe the secret feelings of fear which the constant advance of scientific investigation excites in the minds of many. Is science, then, essentially atheistic and materialistic in its tendency? Does the uniform action of material causes, which we learn with an ever-increasing approximation to certainty, preclude the hypothesis of a benevolent Creator, who has not only designed the existing universe, but who still retains the power to alter its course from time to time?

To enter upon actual theological discussions would be evidently beyond the scope of this work. It is with the scientific method common to all the sciences, and not with any of the separate sciences, that we are concerned. Theology therefore would be at least as much beyond my scope as chemistry or geology. But I believe that grave misapprehensions exist as regards the very nature of scientific method. There are scientific men who assert that the interposition of Providence is impossible, and prayer an absurdity, because the laws of nature are inductively proved to be invariable. Inferences are drawn not so much from particular sciences as from the logical nature of science itself, to negative the impulses and hopes of men. Now I may state that my own studies in logic lead me to call in question such negative inferences. Laws of nature are uniformities observed to exist in the action of certain material agents, but it is logically impossible to show that all other agents must behave as these do. The too exclusive study of particular branches of physical science seems to generate an over-confident and dogmatic spirit. Rejoicing in the success with which a few groups of facts are brought beneath the apparent sway of laws, the investigator hastily assumes that he is close upon the ultimate springs of being. A particle of gelatinous matter is found to obey the ordinary laws of chemistry; yet it moves and lives. The world is therefore asked to believe that chemistry can resolve the mysteries of existence.

*The Meaning of Natural Law.*

Pindar speaks of Law as the Ruler of the Mortals and the Immortals, and it seems to be commonly supposed that the so-called Laws of Nature, in like manner, rule man and his Creator. The course of nature is regarded as being determined by invariable principles of mechanics which have acted since the world began, and will act for evermore. Even if the origin of all things is attributed to an intelligent creative mind, that Being is regarded as having yielded up arbitrary power, and as being subject like a human legislator to the laws which he has himself enacted. Such notions I should describe as superficial and erroneous, being derived, as I think, from false views of the nature of scientific inference, and the degree of certainty of the knowledge which we acquire by inductive investigation.

A law of nature, as I regard the meaning of the expression, is not a uniformity which must be obeyed by all objects, but merely a uniformity which is as a matter of fact obeyed by those objects which have come beneath our observation. There is nothing whatever incompatible with logic in the discovery of objects which should prove exceptions to any law of nature. Perhaps the best established law is that which asserts an invariable correlation to exist between gravity and inertia, so that all gravitating bodies are found to possess inertia, and all bodies possessing inertia are found to gravitate. But it would be no reproach to our scientific method, if something were ultimately discovered to possess gravity without inertia. Strictly defined and correctly interpreted, the law itself would acknowledge the possibility; for with the statement of every law we ought properly to join an estimate of the number of instances in which it has been observed to hold true, and the probability thence calculated, that it will hold true in the next case. Now, as we found (p. 259), no finite number of instances can warrant us in expecting with certainty that the next instance will be of like nature; in the formulas yielded by the inverse method of probabilities a unit always appears to represent the probability that our inference will be mistaken. I demur to the assumption that there is any necessary truth even in such fundamental laws of nature as the Indestructibility of Matter, the Conservation of Energy, or the Laws of Motion. Certain it is that men of science have recognised the conceivability of other laws, and even investigated their mathematical consequences. Airy investigated the mathematical conditions of a perpetual motion (p. 223), and Laplace and Newton discussed imaginary laws of forces inconsistent with those observed to operate in the universe (pp. 642, 706).

The laws of nature, as I venture to regard them, are simply general propositions concerning the correlation of properties which have been observed to hold true of bodies hitherto observed. On the assumption that our experience is of adequate extent, and that no arbitrary interference takes place, we are then able to assign the probability, always less than certainty, that the next object of the same apparent nature will conform to the same laws.

*Infiniteness of the Universe.*

We may safely accept as a satisfactory scientific hypothesis the doctrine so grandly put forth by Laplace, who asserted that a perfect knowledge of the universe, as it existed at any given moment, would give a perfect knowledge of what was to happen thenceforth and for ever after. Scientific inference is impossible, unless we may regard the present as the outcome of what is past, and the cause of what is to come. To the view of perfect intelligence nothing is uncertain. The astronomer can calculate the positions of the heavenly bodies when thousands of generations of men shall have passed away, and in this fact we have some illustration, as Laplace remarks, of the power which scientific prescience may attain. Doubtless, too, all efforts in the investigation of nature tend to bring us nearer to the possession of that ideally perfect power of intelligence. Nevertheless, as Laplace with profound wisdom adds,[603] we must ever remain at an infinite distance from the goal of our aspirations.

[603] *Théorie Analytique des Probabilités*, quoted by Babbage, *Ninth Bridgewater Treatise*, p. 173.

Let us assume, for a time at least, as a highly probable hypothesis, that whatever is to happen must be the outcome of what is; there then arises the question, What is? Now our knowledge of what exists must ever remain imperfect and fallible in two respects. Firstly, we do not know all the matter that has been created, nor the exact manner in which it has been distributed through space. Secondly, assuming that we had that knowledge, we should still be wanting in a perfect knowledge of the way in which the particles of matter will act upon each other. The power of scientific prediction extends at the most to the limits of the data employed. Every conclusion is purely hypothetical and conditional upon the non-interference of agencies previously undetected. The law of gravity asserts that every body tends to approach towards every other body, with a certain determinate force; but, even supposing the law to hold true, it does not assert that the body *will* approach. No single law of nature can warrant us in making an absolute prediction. We must know all the laws of nature and all the existing agents acting according to those laws before we can say what will happen. To assume, then, that scientific method can take everything within its cold embrace of uniformity, is to imply that the Creator cannot outstrip the intelligence of his creatures, and that the existing Universe is not infinite in extent and complexity, an assumption for which I see no logical basis whatever.

*The Indeterminate Problem of Creation.*

A second and very serious misapprehension concerning the import of a law of nature may now be pointed out. It is not uncommonly supposed that a law determines the character of the results which shall take place, as, for instance, that the law of gravity determines what force of gravity shall act upon a given particle. Surely a little reflection must render it plain that a law by itself determines nothing. It is *law plus agents obeying law which has results*, and it is no function of law to govern or define the number and place of its own agents. Whether a particle of matter shall gravitate, depends not only upon the law of Newton, but also upon the distribution of surrounding particles. The theory of gravitation may perhaps be true throughout all time and in all parts of space, and the Creator may never find occasion to create those possible exceptions to it which I have asserted to be conceivable. Let this be as it may; our science cannot certainly determine the question. Certain it is, that the law of gravity does not alone determine the forces which may be brought to bear at any point of space. The force of gravitation acting upon any particle depends upon the mass, distance, and relative position of all the other particles of matter within the bounds of space at the instant in question. Even assuming that all matter when once distributed through space at the Creation was thenceforth to act in an invariable manner without subsequent interference, yet the actual configuration of matter at any moment, and the consequent results of the law of gravitation, must have been entirely a matter of free choice.

Chalmers has most distinctly pointed out that the existing *collocations* of the material world are as important as the laws which the objects obey. He remarks that a certain class of writers entirely overlook the distinction, and forget that mere laws without collocations would have afforded no security against a turbid and disorderly chaos.[604] Mill has recognised[605] the truth of Chalmers’ statement, without drawing the proper inferences from it. He says[606] of the distribution of matter through space, “We can discover nothing regular in the distribution itself; we can reduce it to no uniformity, to no law.” More lately the Duke of Argyll in his well-known work on the *Reign of Law* has drawn attention to the profound distinction between laws and collocations of causes.

[604] *First Bridgewater Treatise* (1834), pp. 16–24.

[605] *System of Logic*, 5th edit. bk. III. chap. V. § 7; chap. XVI. § 3.

[606] *System of Logic*, vol. i. p. 384.

The original conformation of the material universe, as far as we can tell, was free from all restriction. There was unlimited space in which to frame it, and an unlimited number of material particles, each of which could be placed in any one of an infinite number of different positions. It should be added, that each particle might be endowed with any one of an infinite number of quantities of *vis viva* acting in any one of an infinite number of different directions. The problem of Creation was, then, what a mathematician would call *an indeterminate problem*, and it was indeterminate in a great number of ways. Infinitely numerous and various universes might then have been fashioned by the various distribution of the original nebulous matter, although all the particles of matter should obey the law of gravity.

Lucretius tells us how in the original rain of atoms some of these little bodies diverged from the rectilinear direction, and coming into contact with other atoms gave rise to the various combinations of substances which exist. He omitted to tell us whence the atoms came, or by what force some of them were caused to diverge; but surely these omissions involve the whole question. I accept the Lucretian conception of creation when properly supplemented. Every atom which existed in any point of space must have existed there previously, or must have been created there by a previously existing Power. When placed there it must have had a definite mass and a definite energy. Now, as before remarked, an unlimited number of atoms can be placed in unlimited space in an unlimited number of modes of distribution. Out of infinitely infinite choices which were open to the Creator, that one choice must have been made which has yielded the Universe as it now exists.

It would be a mistake, indeed, to suppose that the law of gravity, when it holds true, is no restriction on the distribution of force. That law is a geometrical law, and it would in many cases be mathematically impossible, as far as we can see, that the force of gravity acting on one particle should be small while that on a neighbouring particle is great. We cannot conceive that even Omnipotent Power should make the angles of a triangle greater than two right angles. The primary laws of thought and the fundamental notions of the mathematical sciences do not seem to admit of error or alteration. Into the metaphysical origin and meaning of the apparent necessity attaching to such laws I have not attempted to inquire in this work, and it is not requisite for my present purpose. If the law of gravity were the only law of nature and the Creator had chosen to render all matter obedient to that law, there would doubtless be restrictions upon the effects derivable from any one distribution of matter.

*Hierarchy of Natural Laws.*

A further consideration presents itself. A natural law like that of gravity expresses a certain uniformity in the action of agents submitted to it, and this produces, as we have seen, certain geometrical restrictions upon the effects which those agents may produce. But there are other forces and laws besides gravity. One force may override another, and two laws may each be obeyed and may each disguise the action of the other. In the intimate constitution of matter there may be hidden springs which, while acting in accordance with their own fixed laws, may lead to sudden and unexpected changes. So at least it has been found from time to time in the past, and so there is every reason to believe it will be found in the future. To the ancients it seemed incredible that one lifeless stone could make another leap towards it. A piece of iron while it obeys the magnetic force of the loadstone does not the less obey the law of gravity. A plant gravitates downwards as regards every constituent cell or fibre, and yet it persists in growing upwards. Life is altogether an exception to the simpler phenomena of mineral substances, not in the sense of disproving those laws, but in superadding forces of new and inexplicable character. Doubtless no law of chemistry is broken by the action of the nervous cells, and no law of physics by the pulses of the nervous fibres, but something requires to be added to our sciences in order that we may explain these subtle phenomena.

Now there is absolutely nothing in science or in scientific method to warrant us in assigning a limit to this hierarchy of laws. When in many undoubted cases we find law overriding law, and at certain points in our experience producing unexpected results, we cannot venture to affirm that we have exhausted the strange phenomena which may have been provided for in the original constitution of matter. The Universe might have been so designed that it should go for long intervals through the same round of unvaried existence, and yet that events of exceptional character should be produced from time to time. Babbage showed in that most profound and eloquent work, *The Ninth Bridgewater Treatise*, that it was theoretically possible for human artists to design a machine, consisting of metallic wheels and levers, which should work invariably according to a simple law of action during any finite number of steps, and yet at a fixed moment, however distant, should manifest a single breach of law. Such an engine might go on counting, for instance, the natural numbers until they would reach a number requiring for its expression a hundred million digits. “If every letter in the volume now before the reader’s eyes,” says Babbage,[607] “were changed into a figure, and if all the figures contained in a thousand such volumes were arranged in order, the whole together would yet fall far short of the vast induction the observer would have had in favour of the truth of the law of natural numbers.... Yet shall the engine, true to the prediction of its director, after the lapse of myriads of ages, fulfil its task, and give that one, the first and only exception to that time-sanctioned law. What would have been the chances against the appearance of the excepted case, immediately prior to its occurrence?”

[607] *Ninth Bridgewater Treatise*, p. 140.

As Babbage further showed,[608] a calculating engine, after proceeding through any required number of motions according to a first law, may be made suddenly to suffer a change, so that it shall then commence to calculate according to a wholly new law. After giving the natural numbers for a finite time, it might suddenly begin to give triangular, or square, or cube numbers, and these changes might be conceived theoretically as occurring time after time. Now if such occurrences can be designed and foreseen by a human artist, it is surely within the capacity of the Divine Artist to provide for analogous changes of law in the mechanism of the atom, or the construction of the heavens.

[608] *Ibid.* pp. 34–43.

Physical science, so far as its highest speculations can be trusted, gives some indication of a change of law in the past history of the Universe. According to Sir W. Thomson’s deductions from Fourier’s *Theory of Heat*, we can trace down the dissipation of heat by conduction and radiation to an infinitely distant time when all things will be uniformly cold. But we cannot similarly trace the heat-history of the Universe to an infinite distance in the past. For a certain negative value of the time the formulæ give impossible values, indicating that there was some initial distribution of heat which could not have resulted, according to known laws of nature,[609] from any previous distribution.[610] There are other cases in which a consideration of the dissipation of energy leads to the conception of a limit to the antiquity of the present order of things.[611] Human science, of course, is fallible, and some oversight or erroneous simplification in these theoretical calculations may afterwards be discovered; but as the present state of scientific knowledge is the only ground on which erroneous inferences from the uniformity of nature and the supposed reign of law are founded, I am right in appealing to the present state of science in opposition to these inferences. Now the theory of heat places us in the dilemma either of believing in Creation at an assignable date in the past, or else of supposing that some inexplicable change in the working of natural laws then took place. Physical science gives no countenance to the notion of infinite duration of matter in one continuous course of existence. And if in time past there has been a discontinuity of law, why may there not be a similar event awaiting the world in the future? Infinite ingenuity could have implanted some agency in matter so that it might never yet have made its tremendous powers manifest. We have a very good theory of the conservation of energy, but the foremost physicists do not deny that there may possibly be forms of energy, neither kinetic nor potential, and therefore of unknown nature.[612]

[609] Professor Clifford, in his most interesting lecture on “The First and Last Catastrophe” (*Fortnightly Review*, April 1875, p. 480, reprint by the Sunday Lecture Society, p. 24), objects that I have erroneously substituted “known laws of nature” for “known laws of conduction of heat.” I quite admit the error, without admitting all the conclusions which Professor Clifford proceeds to draw; but I maintain the paragraph unchanged, in order that it may be discussed in the Preface.

[610] Tait’s *Thermodynamics*, p. 38. *Cambridge Mathematical Journal*, vol. iii. p. 174.

[611] Clerk Maxwell’s *Theory of Heat*, p. 245.

[612] Maxwell’s *Theory of Heat*, p. 92.

We can imagine reasoning creatures dwelling in a world where the atmosphere was a mixture of oxygen and inflammable gas like the fire-damp of coal-mines. If devoid of fire, they might have lived through long ages unconscious of the tremendous forces which a single spark would call into play. In the twinkling of an eye new laws might come into action, and the poor reasoning creatures, so confident about their knowledge of the reign of law in their world, would have no time to speculate upon the overthrow of all their theories. Can we with our finite knowledge be sure that such an overthrow of our theories is impossible?

*The Ambiguous Expression, “Uniformity of Nature.”*

I have asserted that serious misconception arises from an erroneous interpretation of the expression Uniformity of Nature. Every law of nature is the statement of a certain uniformity observed to exist among phenomena, and since the laws of nature are invariably obeyed, it seems to follow that the course of nature itself is uniform, so that we can safely judge of the future by the present. This inference is supported by some of the results of physical astronomy. Laplace proved that the planetary system is stable, so that no perturbation which planet produces upon planet can become so great as to cause disruption and permanent alteration of the planetary orbits. A full comprehension of the law of gravity shows that all such disturbances are essentially periodic, so that after the lapse of millions of years the planets will return to the same relative positions, and a new cycle of disturbances will then commence.

As other branches of science progress, we seem to gain assurance that no great alteration of the world’s condition is to be expected. Conflict with a comet has long been the cause of fear, but now it is credibly asserted that we have passed through a comet’s tail without the fact being known at the time, or manifested by any more serious a phenomenon than a slight luminosity of the sky. More recently still the earth is said to have touched the comet Biela, and the only result was a beautiful and perfectly harmless display of meteors. A decrease in the heating power of the sun seems to be the next most probable circumstance from which we might fear the extinction of life on the earth. But calculations founded on reasonable physical data show that no appreciable change can be going on, and experimental data to indicate a change are wholly wanting. Geological investigations show indeed that there have been extensive variations of climate in past times; vast glaciers and icebergs have swept over the temperate regions at one time, and tropical vegetation has flourished near the poles at another time. But here again the vicissitudes of climate assume a periodic character, so that the stability of the earth’s condition does not seem to be threatened.

All these statements may be reasonable, but they do not establish the Uniformity of Nature in the sense that extensive alterations or sudden catastrophes are impossible. In the first place, Laplace’s theory of the stability of the planetary system is of an abstract character, as paying regard to nothing but the mutual gravitation of the planetary bodies and the sun. It overlooks several physical causes of change and decay in the system which were not so well known in his day as at present, and it also presupposes the absence of any interruption of the course of things by conflict with foreign astronomical bodies.

It is now acknowledged by astronomers that there are at least two ways in which the *vis viva* of the planets and satellites may suffer loss. The friction of the tides upon the earth produces a small quantity of heat which is radiated into space, and this loss of energy must result in a decrease of the rotational velocity, so that ultimately the terrestrial day will become identical with the year, just as the periods of revolution of the moon upon its axis and around the earth have already become equal. Secondly, there can be little doubt that certain manifestations of electricity upon the earth’s surface depend upon the relative motions of the planets and the sun, which give rise to periods of increased intensity. Such electrical phenomena must result in the production and dissipation of heat, the energy of which must be drawn, partially at least, from the moving bodies. This effect is probably identical (p. 570) with the loss of energy of comets attributed to the so-called resisting medium. But whatever be the theoretical explanation of these phenomena, it is almost certain that there exists a tendency to the dissipation of the energy of the planetary system, which will, in the indefinite course of time, result in the fall of the planets into the sun.

It is hardly probable, however, that the planetary system will be left undisturbed throughout the enormous interval of time required for the dissipation of its energy in this way. Conflict with other bodies is so far from being improbable, that it becomes approximately certain when we take very long intervals of time into account. As regards cometary conflicts, I am by no means satisfied with the negative conclusions drawn from the remarkable display on the evening of the 27th of November, 1872. We may often have passed through the tail of a comet, the light of which is probably an electrical manifestation no more substantial than the aurora borealis. Every remarkable shower of shooting stars may also be considered as proceeding from a cometary body, so that we may be said to have passed through the thinner parts of innumerable comets. But the earth has probably never passed, in times of which we have any record, through the nucleus of a comet, which consists perhaps of a dense swarm of small meteorites. We can only speculate upon the effects which might be produced by such a conflict, but it would probably be a much more serious event than any yet registered in history. The probability of its occurrence, too, cannot be assigned; for though the probability of conflict with any one cometary nucleus is almost infinitesimal, yet the number of comets is immensely great (p. 408).

It is far from impossible, again, that the planetary system may be invaded by bodies of greater mass than comets. The sun seems to be placed in so extensive a portion of empty space that its own proper motion would not bring it to the nearest known star (α Centauri) in less than 139,200 years. But in order to be sure that this interval of undisturbed life is granted to our globe, we must prove that there are no stars moving so as to meet us, and no dark bodies of considerable size flying through intervening space unknown to us. The intrusion of comets into our system, and the fact that many of them have hyperbolic paths, is sufficient to show that the surrounding parts of space are occupied by multitudes of dark bodies of some size. It is quite probable that small suns may have cooled sufficiently to become non-luminous; for even if we discredit the theory that the variation of brightness of periodic stars is due to the revolution of dark companion stars, yet there is in our own globe an unquestionable example of a smaller body which has cooled below the luminous point.

Altogether, then, it is a mere assumption that the uniformity of nature involves the unaltered existence of our own globe. There is no kind of catastrophe which is too great or too sudden to be theoretically consistent with the reign of law. For all that our science can tell, human history may be closed in the next instant of time. The world may be dashed to pieces against a wandering star; it may be involved in a nebulous atmosphere of hydrogen to be exploded a second afterwards; it may be scorched up or dissipated into vapour by some great explosion in the sun; there might even be within the globe itself some secret cause of disruption, which only needs time for its manifestation.

There are some indications, as already noticed (p. 660), that violent disturbances have actually occurred in the history of the solar system. Olbers sought for the minor planets on the supposition that they were fragments of an exploded planet, and he was rewarded with the discovery of some of them. The retrograde motion of the satellites of the more distant planets, the abnormal position of the poles of Uranus and the excessive distance of Neptune, are other indications of some violent event, of which we have no other evidence. I adduce all these facts and arguments, not to show that there is any considerable probability, as far as we can judge, of interruption within the scope of human history, but to prove that the Uniformity of Nature is theoretically consistent with the most unexpected events of which we can form a conception.

*Possible States of the Universe.*

When we give the rein to scientific imagination, it becomes apparent that conflict of body with body must not be regarded as the rare exception, but as the general rule and the inevitable fate of each star system. So far as we can trace out the results of the law of gravitation, and of the dissipation of energy, the universe must be regarded as undergoing gradual condensation into a single cold solid body of gigantic dimensions. Those who so frequently use the expression Uniformity of Nature seem to forget that the Universe might exist consistently with the laws of nature in the most diverse conditions. It might consist, on the one hand, of a glowing nebulous mass of gaseous substances. The heat might be so intense that all elements, even carbon and silicon, would be in the state of gas, and all atoms, of whatever nature, would be flying about in chemical independence, diffusing themselves almost uniformly in the neighbouring parts of space. There would then be no life, unless we can apply that name to the passage through each part of space of similar average trains of atoms, the particular succession of atoms being governed only by the theory of probability, and the law of divergence from a mean exhibited in the Arithmetical Triangle. Such a universe would correspond partially to the Lucretian rain of atoms, and to that nebular hypothesis out of which Laplace proposed philosophically to explain the evolution of the planetary system.

According to another extreme supposition, the intense heat-energy of this nebulous mass might be radiated away into the unknown regions of outer space. The attraction of gravity would exert itself between each two particles, and the energy of motion thence arising would, by incessant conflicts, be resolved into heat and dissipated. Inconceivable ages might be required for the completion of this process, but the dissipation of energy thus proceeding could end only in the production of a cold and motionless universe. The relation of cause and effect, as we see it manifested in life and growth, would degenerate into the constant existence of every particle in a fixed position relative to every other particle. Logical and geometrical resemblances would still exist between atoms, and between groups of atoms crystallised in their appropriate forms for evermore. But time, the great variable, would bring no variation, and as to human hopes and troubles, they would have gone to eternal rest.

Science is not really adequate to proving that such is the inevitable fate of the universe, for we can seldom trust our best-established theories far from their data. Nevertheless, the most probable speculations which we can form as to the history, especially of our own planetary system, is that it originated in a heated revolving nebulous mass of gas, and is in a state of excessively slow progress towards the cold and stony condition. Other speculative hypotheses might doubtless be entertained. Every hypothesis is pressed by difficulties. If the whole universe be cooling, whither does the heat go? If we are to get rid of it entirely, outer space must be infinite in extent, so that it shall never be stopped and reflected back. But not to speak of metaphysical difficulties, if the medium of heat undulations be infinite in extent, why should not the material bodies placed in it be infinite also in number and aggregate mass? It is apparent that we are venturing into speculations which surpass our powers of scientific inference. But then I am arguing negatively; I wish to show that those who speak of the uniformity of nature, and the reign of law, misinterpret the meaning involved in those expressions. Law is not inconsistent with extreme diversity, and, so far as we can read the history of this planetary system, it did probably originate in heated nebulous matter, and man’s history forms but a brief span in its progress towards the cold and stony condition. It is by doubtful and speculative hypotheses alone that we can avoid such a conclusion, and I depart least from undoubted facts and well-established laws when I assert that, whatever uniformities may underlie the phenomena of nature, constant variety and ever-progressing change is the real outcome.

*Speculations on the Reconcentration of Energy.*

There are unequivocal indications, as I have said, that the material universe, as we at present see it, is progressing from some act of creation, or some discontinuity of existence of which the date may be approximately fixed by scientific inference. It is progressing towards a state in which the available energy of matter will be dissipated through infinite surrounding space, and all matter will become cold and lifeless. This constitutes, as it were, the historical period of physical science, that over which our scientific foresight may more or less extend. But in this, as in other cases, we have no right to interpret our experience negatively, so as to infer that because the present state of things began at a particular time, there was no previous existence. It may be that the present period of material existence is but one of an indefinite series of like periods. All that we can see, and feel, and infer, and reason about may be, as it were, but a part of one single pulsation in the existence of the universe.

After Sir W. Thomson had pointed out the preponderating tendency which now seems to exist towards the conversion of all energy into heat-energy, and its equal diffusion by radiation throughout space, the late Professor Rankine put forth a remarkable speculation.[613] He suggested that the ethereal, or, as I have called it, the *adamantine* medium in which all the stars exist, and all radiation takes place, may have bounds, beyond which only empty space exists. All heat undulations reaching this boundary will be totally reflected, according to the theory of undulations, and will be reconcentrated into foci situated in various parts of the medium. Whenever a cold and extinct star happens to pass through one of these foci, it will be instantly ignited and resolved by intense heat into its constituent elements. Discontinuity will occur in the history of that portion of matter, and the star will begin its history afresh with a renewed store of energy.

[613] *Report of the British Association* (1852), Report of Sections, p. 12.

This is doubtless a mere speculation, practically incapable of verification by observation, and almost free from restrictions afforded by present knowledge. We might attribute various shapes to the adamantine medium, and the consequences would be various. But there is this value in such speculations, that they draw attention to the finiteness of our knowledge. We cannot deny the possible truth of such an hypothesis, nor can we place a limit to the scientific imagination in the framing of other like hypotheses. It is impossible, indeed, to follow out our scientific inferences without falling into speculation. If heat be radiated into outward space, it must either proceed *ad infinitum*, or it must be stopped somewhere. In the latter case we fall upon Rankine’s hypothesis. But if the material universe consist of a finite collection of heated matter situated in a finite portion of an infinite adamantine medium, then either this universe must have existed for a finite time, or else it must have cooled down during the infinity of past time indefinitely near to the absolute zero of temperature. I objected to Lucretius’ argument against the destructibility of matter, that we have no knowledge whatever of the laws according to which it would undergo destruction. But we do know the laws according to which the dissipation of heat appears to proceed, and the conclusion inevitably is that a finite heated material body placed in a perfectly cold infinitely extended medium would in an infinite time sink to zero of temperature. Now our own world is not yet cooled down near to zero, so that physical science seems to place us in the dilemma of admitting either the finiteness of past duration of the world, or else the finiteness of the portion of medium in which we exist. In either case we become involved in metaphysical and mechanical difficulties surpassing our mental powers.

*The Divergent Scope for New Discovery.*

In the writings of some recent philosophers, especially of Auguste Comte, and in some degree John Stuart Mill, there is an erroneous and hurtful tendency to represent our knowledge as assuming an approximately complete character. At least these and many other writers fail to impress upon their readers a truth which cannot be too constantly borne in mind, namely, that the utmost successes which our scientific method can accomplish will not enable us to comprehend more than an infinitesimal fraction of what there doubtless is to comprehend.[614] Professor Tyndall seems to me open to the same charge in a less degree. He remarks[615] that we can probably never bring natural phenomena completely under mathematical laws, because the approach of our sciences towards completeness may be asymptotic, so that however far we may go, there may still remain some facts not subject to scientific explanation. He thus likens the supply of novel phenomena to a convergent series, the earlier and larger terms of which have been successfully disposed of, so that comparatively minor groups of phenomena alone remain for future investigators to occupy themselves upon.

[614] Mr. C. J. Monroe objects that in this statement I do injustice to Comte, who, he thinks, did impress upon his readers the inadequacy of our mental powers compared with the vastness of the subject matter of science. The error of Comte, he holds, was in maintaining that science had been carried about as far as it is worth while to carry it, which is a different matter. In either case, Comte’s position is so untenable that I am content to leave the question undecided.

[615] *Fragments of Science*, p. 362.

On the contrary, as it appears to me, the supply of new and unexplained facts is divergent in extent, so that the more we have explained, the more there is to explain. The further we advance in any generalisation, the more numerous and intricate are the exceptional cases still demanding further treatment. The experiments of Boyle, Mariotte, Dalton, Gay-Lussac, and others, upon the physical properties of gases, might seem to have exhausted that subject by showing that all gases obey the same laws as regards temperature, pressure, and volume. But in reality these laws are only approximately true, and the divergences afford a wide and quite unexhausted field for further generalisation. The recent discoveries of Professor Andrews have summed up some of these exceptional facts under a wider generalisation, but in reality they have opened to us vast new regions of interesting inquiry, and they leave wholly untouched the question why one gas behaves differently from another.

The science of crystallography is that perhaps in which the most precise and general laws have been detected, but it would be untrue to assert that it has lessened the area of future discovery. We can show that each one of the seven or eight hundred forms of calcite is derivable by geometrical modifications from an hexagonal prism; but who has attempted to explain the molecular forces producing these modifications, or the chemical conditions in which they arise? The law of isomorphism is an important generalisation, for it establishes a general resemblance between the forms of crystallisation of natural classes of elements. But if we examine a little more closely we find that these forms are only approximately alike, and the divergence peculiar to each substance is an unexplained exception.

By many similar illustrations it might readily be shown that in whatever direction we extend our investigations and successfully harmonise a few facts, the result is only to raise up a host of other unexplained facts. Can any scientific man venture to state that there is less opening now for new discoveries than there was three centuries ago? Is it not rather true that we have but to open a scientific book and read a page or two, and we shall come to some recorded phenomenon of which no explanation can yet be given? In every such fact there is a possible opening for new discoveries, and it can only be the fault of the investigator’s mind if he can look around him and find no scope for the exercise of his faculties.

*Infinite Incompleteness of the Mathematical Sciences.*

There is one privilege which a certain amount of knowledge should confer; it is that of becoming aware of the weakness of our powers compared with the tasks which they might undertake if stronger. To the poor savage who cannot count twenty the arithmetical accomplishments of the schoolboy are miraculously great. The schoolboy cannot comprehend the vastly greater powers of the student, who has acquired facility in algebraic processes. The student can but look with feelings of surprise and reverence at the powers of a Newton or a Laplace. But the question at once suggests itself, Do the powers of the highest human intellect bear a finite ratio to the things which are to be understood and calculated? How many further steps must we take in the rise of mental ability and the extension of mathematical methods before we begin to exhaust the knowable?

I am inclined to find fault with mathematical writers because they often exult in what they can accomplish, and omit to point out that what they do is but an infinitely small part of what might be done. They exhibit a general inclination, with few exceptions, not to do so much as mention the existence of problems of an impracticable character. This may be excusable as far as the immediate practical result of their researches is in question, but the custom has the effect of misleading the general public into the fallacious notion that mathematics is a *perfect* science, which accomplishes what it undertakes in a complete manner. On the contrary, it may be said that if a mathematical problem were selected by chance out of the whole number which might be proposed, the probability is infinitely slight that a human mathematician could solve it. Just as the numbers we can count are nothing compared with the numbers which might exist, so the accomplishments of a Laplace or a Lagrange are, as it were, the little corner of the multiplication-table, which has really an infinite extent.

I have pointed out that the rude character of our observations prevents us from being aware of the greater number of effects and actions in nature. It must be added that, if we perceive them, we should usually be incapable of including them in our theories from want of mathematical power. Some persons may be surprised that though nearly two centuries have elapsed since the time of Newton’s discoveries, we have yet no general theory of molecular action. Some approximations have been made towards such a theory. Joule and Clausius have measured the velocity of gaseous atoms, or even determined the average distance between the collisions of atom and atom. Thomson has approximated to the number of atoms in a given bulk of substance. Rankine has formed some reasonable hypotheses as to the actual constitution of atoms. It would be a mistake to suppose that these ingenious results of theory and experiment form any appreciable approach to a complete solution of molecular motions. There is every reason to believe, judging from the spectra of the elements, their atomic weights and other data, that chemical atoms are very complicated structures. An atom of pure iron is probably a far more complicated system than that of the planets and their satellites. A compound atom may perhaps be compared with a stellar system, each star a minor system in itself. The smallest particle of solid substance will consist of a great number of such stellar systems united in regular order, each bounded by the other, communicating with it in some manner yet wholly incomprehensible. What are our mathematical powers in comparison with this problem?

After two centuries of continuous labour, the most gifted men have succeeded in calculating the mutual effects of three bodies each upon the other, under the simple hypothesis of the law of gravity. Concerning these calculations we must further remember that they are purely approximate, and that the methods would not apply where four or more bodies are acting, and all produce considerable effects upon each other. There is reason to believe that each constituent of a chemical atom goes through an orbit in the millionth part of the twinkling of an eye. In each revolution it is successively or simultaneously under the influence of many other constituents, or possibly comes into collision with them. It is no exaggeration to say that mathematicians have the least notion of the way in which they could successfully attack so difficult a problem of forces and motions. As Herschel has remarked,[616] each of these particles is for ever solving differential equations, which, if written out in full, might belt the earth.

[616] *Familiar Lectures on Scientific Subjects*, p. 458.

Some of the most extensive calculations ever made were those required for the reduction of the measurements executed in the course of the Trigonometrical Survey of Great Britain. The calculations arising out of the principal triangulation occupied twenty calculators during three or four years, in the course of which the computers had to solve simultaneous equations involving seventy-seven unknown quantities. The reduction of the levellings required the solution of a system of ninety-one equations. But these vast calculations present no approach whatever to what would be requisite for the complete treatment of any one physical problem. The motion of glaciers is supposed to be moderately well understood in the present day. A glacier is a viscid, slowly yielding mass, neither absolutely solid nor absolutely rigid, but it is expressly remarked by Forbes,[617] that not even an approximate solution of the mathematical conditions of such a moving mass can yet be possible. “Every one knows,” he says, “that such problems are beyond the compass of exact mathematics;” but though mathematicians may know this, they do not often enough impress that knowledge on other people.

[617] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 406.

The problems which are solved in our mathematical books consist of a small selection of those which happen from peculiar conditions to be solvable. But the very simplest problem in appearance will often give rise to impracticable calculations. Mr. Todhunter[618] seems to blame Condorcet, because in one of his memoirs he mentions a problem to solve which would require a great and impracticable number of successive integrations. Now, if our mathematical sciences are to cope with the problems which await solution, we must be prepared to effect an unlimited number of successive integrations; yet at present, and almost beyond doubt for ever, the probability that an integration taken haphazard will come within our powers is exceedingly small.

[618] *History of the Theory of Probability*, p. 398.

In some passages of that remarkable work, the *Ninth Bridgewater Treatise* (pp. 113–115), Babbage has pointed out that if we had power to follow and detect the minutest effects of any disturbance, each particle of existing matter would furnish a register of all that has happened. “The track of every canoe--of every vessel that has yet disturbed the surface of the ocean, whether impelled by manual force or elemental power, remains for ever registered in the future movement of all succeeding particles which may occupy its place. The furrow which it left is, indeed, instantly filled up by the closing waters; but they draw after them other and larger portions of the surrounding element, and these again, once moved, communicate motion to others in endless succession.” We may even say that “The air itself is one vast library, on whose pages are for ever written all that man has ever said or even whispered. There, in their mutable but unerring characters, mixed with the earliest as well as the latest sighs of mortality, stand for ever recorded, vows unredeemed, promises unfulfilled, perpetuating in the united movements of each particle the testimony of man’s changeful will.”

When we read reflections such as these, we may congratulate ourselves that we have been endowed with minds which, rightly employed, can form some estimate of their incapacity to trace out and account for all that proceeds in the simpler actions of material nature. It ought to be added that, wonderful as is the extent of physical phenomena open to our investigation, intellectual phenomena are yet vastly more extensive. Of this I might present one satisfactory proof were space available by pointing out that the mathematical functions employed in the calculations of physical science form an infinitely small fraction of the functions which might be invented. Common trigonometry consists of a great series of useful formulæ, all of which arise out of the relation of the sine and cosine expressed in one equation, sin ^{2}*x* + cos ^{2}*x* = 1. But this is not the only trigonometry which may exist; mathematicians also recognise hyperbolic trigonometry, of which the fundamental equation is cos ^{2}*x* - sin ^{2}*x* = 1. De Morgan has pointed out that the symbols of ordinary algebra form but three of an interminable series of conceivable systems.[619] As the logarithmic operation is to addition or addition to multiplication, so is the latter to a higher operation, and so on without limit.

[619] *Trigonometry and Double Algebra*, chap. ix.

We may rely upon it that immense, and to us inconceivable, advances will be made by the human intellect, in the absence of any catastrophe to the species or the globe. Within historical periods we can trace the rise of mathematical science from its simplest germs. We can prove our descent from ancestors who counted only on their fingers. How infinitely is a Newton or a Laplace above those simple savages. Pythagoras is said to have sacrificed a hecatomb when he discovered the forty-seventh proposition of Euclid, and the occasion was worthy of the sacrifice. Archimedes was beside himself when he first perceived his beautiful mode of determining specific gravities. Yet these great discoveries are the commonplaces of our school books. Step by step we can trace upwards the acquirement of new mental powers. What could be more wonderful than Napier’s discovery of logarithms, a new mode of calculation which has multiplied perhaps a hundredfold the working powers of every computer, and has rendered easy calculations which were before impracticable? Since the time of Newton and Leibnitz worlds of problems have been solved which before were hardly conceived as matters of inquiry. In our own day extended methods of mathematical reasoning, such as the system of quaternions, have been brought into existence. What intelligent man will doubt that the recondite speculations of a Cayley, a Sylvester, or a Clifford may lead to some new development of new mathematical power, at the simplicity of which a future age will wonder, and yet wonder more that to us they were so dark and difficult. May we not repeat the words of Seneca: “Veniet tempus, quo ista quæ nunc latent, in lucem dies extrahat, et longioris ævi diligentia: ad inquisitionem tantorum ætas una non sufficit. Veniet tempus, quo posteri nostri tam aperta nos nescisse mirentur.”

*The Reign of Law in Mental and Social Phenomena.*

After we pass from the so-called physical sciences to those which attempt to investigate mental and social phenomena, the same general conclusions will hold true. No one will be found to deny that there are certain uniformities of thinking and acting which can be detected in reasoning beings, and so far as we detect such laws we successfully apply scientific method. But those who attempt to establish social or moral sciences soon become aware that they are dealing with subjects of enormous perplexity. Take as an instance the science of political economy. If a science at all, it must be a mathematical science, because it deals with quantities of commodities. But as soon as we attempt to draw out the equations expressing the laws of demand and supply, we discover that they have a complexity entirely surpassing our powers of mathematical treatment. We may lay down the general form of the equations, expressing the demand and supply for two or three commodities among two or three trading bodies, but all the functions involved are so complicated in character that there is not much fear of scientific method making rapid progress in this direction. If such be the prospects of a comparatively formal science, like political economy, what shall we say of moral science? Any complete theory of morals must deal with quantities of pleasure and pain, as Bentham pointed out, and must sum up the general tendency of each kind of action upon the good of the community. If we are to apply scientific method to morals, we must have a calculus of moral effects, a kind of physical astronomy investigating the mutual perturbations of individuals. But as astronomers have not yet fully solved the problem of three gravitating bodies, when shall we have a solution of the problem of three moral bodies?

The sciences of political economy and morality are comparatively abstract and general, treating mankind from simple points of view, and attempting to detect general principles of action. They are to social phenomena what the abstract sciences of chemistry, heat, and electricity are to the concrete science of meteorology. Before we can investigate the actions of any aggregate of men, we must have fairly mastered all the more abstract sciences applying to them, somewhat in the way that we have acquired a fair comprehension of the simpler truths of chemistry and physics. But all our physical sciences do not enable us to predict the weather two days hence with any great probability, and the general problem of meteorology is almost unattempted as yet. What shall we say then of the general problem of social science, which shall enable us to predict the course of events in a nation?

Several writers have proposed to lay the foundations of the science of history. Buckle undertook to write the *History of Civilisation in England*, and to show how the character of a nation could be explained by the nature of the climate and the fertility of the soil. He omitted to explain the contrast between the ancient Greek nation and the present one; there must have been an extraordinary revolution in the climate or the soil. Auguste Comte detected the simple laws of the course of development through which nations pass. There are always three phases of intellectual condition,--the theological, the metaphysical, and the positive; applying this general law of progress to concrete cases, Comte was enabled to predict that in the hierarchy of European nations, Spain would necessarily hold the highest place. Such are the parodies of science offered to us by the *positive* philosophers.

A science of history in the true sense of the term is an absurd notion. A nation is not a mere sum of individuals whom we can treat by the method of averages; it is an organic whole, held together by ties of infinite complexity. Each individual acts and re-acts upon his smaller or greater circle of friends, and those who acquire a public position exert an influence on much larger sections of the nation. There will always be a few great leaders of exceptional genius or opportunities, the unaccountable phases of whose opinions and inclinations sway the whole body. From time to time arise critical situations, battles, delicate negotiations, internal disturbances, in which the slightest incidents may change the course of history. A rainy day may hinder a forced march, and change the course of a campaign; a few injudicious words in a despatch may irritate the national pride; the accidental discharge of a gun may precipitate a collision the effects of which will last for centuries. It is said that the history of Europe depended at one moment upon the question whether the look-out man upon Nelson’s vessel would or would not descry a ship of Napoleon’s expedition to Egypt which was passing not far off. In human affairs, then, the smallest causes may produce the greatest effects, and the real application of scientific method is out of the question.

*The Theory of Evolution.*

Profound philosophers have lately generalised concerning the production of living forms and the mental and moral phenomena regarded as their highest development. Herbert Spencer’s theory of evolution purports to explain the origin of all specific differences, so that not even the rise of a Homer or a Beethoven would escape from his broad theories. The homogeneous is unstable and must differentiate itself, says Spencer, and hence comes the variety of human institutions and characters. In order that a living form shall continue to exist and propagate its kind, says Darwin, it must be suitable to its circumstances, and the most suitable forms will prevail over and extirpate those which are less suitable. From these fruitful ideas are developed theories of evolution and natural selection which go far towards accounting for the existence of immense numbers of living creatures--plants, and animals. Apparent adaptations of organs to useful purposes, which Paley regarded as distinct products of creative intelligence, are now seen to follow as natural effects of a constantly acting tendency. Even man, according to these theories, is no distinct creation, but rather an extreme case of brain development. His nearest cousins are the apes, and his pedigree extends backwards until it joins that of the lowliest zoophytes.

The theories of Darwin and Spencer are doubtless not demonstrated; they are to some extent hypothetical, just as all the theories of physical science are to some extent hypothetical, and open to doubt. Judging from the immense numbers of diverse facts which they harmonise and explain, I venture to look upon the theories of evolution and natural selection in their main features as two of the most probable hypotheses ever proposed. I question whether any scientific works which have appeared since the *Principia* of Newton are comparable in importance with those of Darwin and Spencer, revolutionising as they do all our views of the origin of bodily, mental, moral, and social phenomena.

Granting all this, I cannot for a moment admit that the theory of evolution will destroy theology. That theory embraces several laws or uniformities which are observed to be true in the production of living forms; but these laws do not determine the size and figure of living creatures, any more than the law of gravitation determines the magnitudes and distances of the planets. Suppose that Darwin is correct in saying that man is descended from the Ascidians: yet the precise form of the human body must have been influenced by an infinite train of circumstances affecting the reproduction, growth, and health of the whole chain of intermediate beings. No doubt, the circumstances being what they were, man could not be otherwise than he is, and if in any other part of the universe an exactly similar earth, furnished with exactly similar germs of life, existed, a race must have grown up there exactly similar to the human race.

By a different distribution of atoms in the primeval world a different series of living forms on this earth would have been produced. From the same causes acting according to the same laws, the same results will follow; but from different causes acting according to the same laws, different results will follow. So far as we can see, then, infinitely diverse living creatures might have been created consistently with the theory of evolution, and the precise reason why we have a backbone, two hands with opposable thumbs, an erect stature, a complex brain, about 223 bones, and many other peculiarities, is only to be found in the original act of creation. I do not, any less than Paley, believe that the eye of man manifests design. I believe that the eye was gradually developed, and we can in fact trace its gradual development from the first germ of a nerve affected by light-rays in some simple zoophyte. In proportion as the eye became a more accurate instrument of vision, it enabled its possessor the better to escape destruction, but the ultimate result must have been contained in the aggregate of the causes, and these causes, as far as we can see, were subject to the arbitrary choice of the Creator.

Although Agassiz was clearly wrong in holding that every species of living creature appeared on earth by the immediate intervention of the Creator, which would amount to saying that no laws of connection between forms are discoverable, yet he seems to be right in asserting that living forms are distinct from those produced by purely physical causes. “The products of what are commonly called physical agents,” he says,[620] “are everywhere the same (*i.e.* upon the whole surface of the earth), and have always been the same (*i.e.* during all geological periods); while organised beings are everywhere different and have differed in all ages. Between two such series of phenomena there can be no causal or genetic connection.” Living forms as we now regard them are essentially variable, but from constant mechanical causes constant effects would ensue. If vegetable cells are formed on geometrical principles being first spherical, and then by mutual compression dodecahedral, then all cells should have similar forms. In the Foraminifera and some other lowly organisms, we seem to observe the production of complex forms on geometrical principles. But from similar causes acting according to similar laws only similar results could be produced. If the original life germ of each creature is a simple particle of protoplasm, unendowed with any distinctive forces, then the whole of the complex phenomena of animal and vegetable life are effects without causes. Protoplasm may be chemically the same substance, and the germ-cell of a man and of a fish may be apparently the same, so far as the microscope can decide; but if certain cells produce men, and others as uniformly produce a species of fish, there must be a hidden constitution determining the extremely different results. If this were not so, the generation of every living creature from the uniform germ would have to be regarded as a distinct act of creation.

[620] Agassiz, *Essay on Classification*, p. 75.

Theologians have dreaded the establishment of the theories of Darwin and Huxley and Spencer, as if they thought that those theories could explain everything upon the purest mechanical and material principles, and exclude all notions of design. They do not see that those theories have opened up more questions than they have closed. The doctrine of evolution gives a complete explanation of no single living form. While showing the general principles which prevail in the variation of living creatures, it only points out the infinite complexity of the causes and circumstances which have led to the present state of things. Any one of Mr. Darwin’s books, admirable though they all are, consists but in the setting forth of a multitude of indeterminate problems. He proves in the most beautiful manner that each flower of an orchid is adapted to some insect which frequents and fertilises it, and these adaptations are but a few cases of those immensely numerous ones which have occurred in the lives of plants and animals. But why orchids should have been formed so differently from other plants, why anything, indeed, should be as it is, rather than in some of the other infinitely numerous possible modes of existence, he can never show. The origin of everything that exists is wrapped up in the past history of the universe. At some one or more points in past time there must have been arbitrary determinations which led to the production of things as they are.

*Possibility of Divine Interference.*

I will now draw the reader’s attention to pages 149 to 152. I there pointed out that all inductive inference involves the assumption that our knowledge of what exists is complete, and that the conditions of things remain unaltered between the time of our experience and the time to which our inferences refer. Recurring to the illustration of a ballot-box, employed in the chapter on the inverse method of probabilities, we assume when predicting the probable nature of the next drawing, firstly, that our previous drawings have been sufficiently numerous to give us knowledge of the contents of the box; and, secondly, that no interference with the ballot-box takes place between the previous and the next drawings. The results yielded by the theory of probability are quite plain. No finite number of casual drawings can give us sure knowledge of the contents of the box, so that, even in the absence of all disturbance, our inferences are merely the best which can be made, and do not approach to infallibility. If, however, interference be possible, even the theory of probability ceases to be applicable, for, the amount and nature of that interference being arbitrary and unknown, there ceases to be any connection between premises and conclusion. Many years of reflection have not enabled me to see the way of avoiding this hiatus in scientific certainty. The conclusions of scientific inference appear to be always of a hypothetical and provisional nature. Given certain experience, the theory of probability yields us the true interpretation of that experience and is the surest guide open to us. But the best calculated results which it can give are never absolute probabilities; they are purely relative to the extent of our information. It seems to be impossible for us to judge how far our experience gives us adequate information of the universe as a whole, and of all the forces and phenomena which can have place therein.

I feel that I cannot in the space remaining at my command in the present volume, sufficiently follow out the lines of thought suggested, or define with precision my own conclusions. This chapter contains merely *Reflections* upon subjects of so weighty a character that I should myself wish for many years--nay for more than a lifetime of further reflection. My purpose, as I have repeatedly said, is the purely negative one of showing that atheism and materialism are no necessary results of scientific method. From the preceding reviews of the value of our scientific knowledge, I draw one distinct conclusion, that we cannot disprove the possibility of Divine interference in the course of nature. Such interference might arise, so far as our knowledge extends, in two ways. It might consist in the disclosure of the existence of some agent or spring of energy previously unknown, but which effects a given purpose at a given moment. Like the pre-arranged change of law in Babbage’s imaginary calculating machine, there may exist pre-arranged surprises in the order of nature, as it presents itself to us. Secondly, the same Power, which created material nature, might, so far as I can see, create additions to it, or annihilate portions which do exist. Such events are in a certain sense inconceivable to us; yet they are no more inconceivable than the existence of the world as it is. The indestructibility of matter, and the conservation of energy, are very probable scientific hypotheses, which accord satisfactorily with experiments of scientific men during a few years past, but it would be gross misconception of scientific inference to suppose that they are certain in the sense that a proposition in geometry is certain. Philosophers no doubt hold that *de nihilo nihil fit*, that is to say, their senses give them no means of imagining to the mind how creation can take place. But we are on the horns of a trilemma; we must either deny that anything exists, or we must allow that it was created out of nothing at some moment of past time, or that it existed from eternity. The first alternative is absurd; the other two seem to me equally conceivable.

*Conclusion.*

It may seem that there is one point where our speculations must end, namely where contradiction begins. The laws of Identity and Difference and Duality were the foundations from which we started, and they are, so far as I can see, the foundations which we can never quit without tottering. Scientific Method must begin and end with the laws of thought, but it does not follow that it will save us from encountering inexplicable, and at least apparently contradictory results. The nature of continuous quantity leads us into extreme difficulties. Any finite space is composed of an infinite number of infinitely small spaces, each of which, again, is composed of an infinite number of spaces of a second order of smallness; these spaces of the second order are composed, again, of infinitely small spaces of the third order. Even these spaces of the third order are not absolute geometrical points answering to Euclid’s definition of a point, as position without magnitude. Go on as far as we will, in the subdivision of continuous quantity, yet we never get down to the absolute point. Thus scientific method leads us to the inevitable conception of an infinite series of successive orders of infinitely small quantities. If so, there is nothing impossible in the existence of a myriad universes within the compass of a needle’s point, each with its stellar systems, and its suns and planets, in number and variety unlimited. Science does nothing to reduce the number of strange things that we may believe. When fairly pursued it makes absurd drafts upon our powers of comprehension and belief.

Some of the most precise and beautiful theorems in mathematical science seem to me to involve apparent contradiction. Can we imagine that a point moving along a perfectly straight line towards the west would ever get round to the east and come back again, having performed, as it were, a circuit through infinite space, yet without ever diverging from a perfectly straight direction? Yet this is what happens to the intersecting point of two straight lines in the same plane, when one line revolves. The same paradox is exhibited in the hyperbola regarded as an infinite ellipse, one extremity of which has passed to an infinite distance and come back in the opposite direction. A varying quantity may change its sign by passing either through zero or through infinity. In the latter case there must be one intermediate value of the variable for which the variant is indifferently negative infinity and positive infinity. Professor Clifford tells me that he has found a mathematical function which approaches infinity as the variable approaches a certain limit; yet at the limit the function is finite! Mathematicians may shirk difficulties, but they cannot make such results of mathematical principles appear otherwise than contradictory to our common notions of space.

The hypothesis that there is a Creator at once all-powerful and all-benevolent is pressed, as it must seem to every candid investigator, with difficulties verging closely upon logical contradiction. The existence of the smallest amount of pain and evil would seem to show that He is either not perfectly benevolent, or not all-powerful. No one can have lived long without experiencing sorrowful events of which the significance is inexplicable. But if we cannot succeed in avoiding contradiction in our notions of elementary geometry, can we expect that the ultimate purposes of existence shall present themselves to us with perfect clearness? I can see nothing to forbid the notion that in a higher state of intelligence much that is now obscure may become clear. We perpetually find ourselves in the position of finite minds attempting infinite problems, and can we be sure that where we see contradiction, an infinite intelligence might not discover perfect logical harmony?

From science, modestly pursued, with a due consciousness of the extreme finitude of our intellectual powers, there can arise only nobler and wider notions of the purpose of Creation. Our philosophy will be an affirmative one, not the false and negative dogmas of Auguste Comte, which have usurped the name, and misrepresented the tendencies of a true *positive philosophy*. True science will not deny the existence of things because they cannot be weighed and measured. It will rather lead us to believe that the wonders and subtleties of possible existence surpass all that our mental powers allow us clearly to perceive. The study of logical and mathematical forms has convinced me that even space itself is no requisite condition of conceivable existence. Everything, we are told by materialists, must be here or there, nearer or further, before or after. I deny this, and point to logical relations as my proof.

There formerly seemed to me to be something mysterious in the denominators of the binomial expansion (p. 190), which are reproduced in the natural constant ε, or

1 + 1/1 + 1/(1 . 2) + 1/(1 . 2 . 3) + ...

and in many results of mathematical analysis. I now perceive, as already explained (pp. 33, 160, 383), that they arise out of the fact that the relations of space do not apply to the logical conditions governing the numbers of combinations as contrasted to those of permutations. So far am I from accepting Kant’s doctrine that space is a necessary form of thought, that I regard it as an accident, and an impediment to pure logical reasoning. Material existences must exist in space, no doubt, but intellectual existences may be neither in space nor out of space; they may have no relation to space at all, just as space itself has no relation to time. For all that I can see, then, there may be intellectual existences to which both time and space are nullities.

Now among the most unquestionable rules of scientific method is that first law that *whatever phenomenon is, is*. We must ignore no existence whatever; we may variously interpret or explain its meaning and origin, but, if a phenomenon does exist, it demands some kind of explanation. If then there is to be competition for scientific recognition, the world without us must yield to the undoubted existence of the spirit within. Our own hopes and wishes and determinations are the most undoubted phenomena within the sphere of consciousness. If men do act, feel, and live as if they were not merely the brief products of a casual conjunction of atoms, but the instruments of a far-reaching purpose, are we to record all other phenomena and pass over these? We investigate the instincts of the ant and the bee and the beaver, and discover that they are led by an inscrutable agency to work towards a distant purpose. Let us be faithful to our scientific method, and investigate also those instincts of the human mind by which man is led to work as if the approval of a Higher Being were the aim of life.

INDEX.

Abacus, logical, 104; arithmetical, 107; Panchrestus, 182.

Aberration of light, 561; systematic, 547.

Abscissio infiniti, 79, 713.

Abstract terms, 27; number, 159.

Abstraction, 704; logical, 25; numerical, 158; of indifferent circumstances, 97.

Accademia del Cimento, 427, 432, 436, 527.

Accident, logical, 700.

Accidental discovery, 529.

Achromatic lenses, 432.

Actinometer, 337.

Adamantine medium, 605, 751.

Adjectives, 14, 30, 31, 35; indeterminate, 41.

Adrain, of New Brunswick, 375.

Affirmation, 44.

Agassiz, on genera, 726; on creation of species, 763.

Agreement, 44.

Airy, Sir George Biddell, on perpetual motion, 223; new property of sphere, 232; pendulum experiments, 291, 304, 348, 567; standard clock, 353; book on *Errors of Observation*, 395; tides, 488; extra-polation, 495; Thales’ eclipse, 537; interference of light, 539; density of earth, 291.

Alchemists, 505; how misled, 428.

Algebra, 123, 155, 164; Diophantine, 631.

Algebraic, equations, 123; geometry, 633.

Allotropic state, 663, 670.

Alloys, possible number, 191; properties, 528.

Alphabet, the Logical, 93, 104, 125; Morse, 193.

Alphabet, permutations of letters of the, 174, 179.

Alphabetic indexes, 714.

Alternative relations, 67; exclusive and unexclusive, 205.

Ampère, electricity, 547; classification, 679.

Anagrams, 128.

Analogy, 627; of logical and numerical terms, 160; and generalisation, 596; in mathematical sciences, 631; in theory of undulations, 635; in astronomy, 638; failure of, 641.

Analysis, logical, 122.

Andrews, Prof. Thomas, experiments on gaseous state, 71, 613, 665, 753.

Angström, on spectrum, 424.

Angular magnitude, 305, 306, 326.

Antecedent defined, 225.

Anticipation of Nature, 509.

Anticipations, of Principle of Substitution, 21; of electric telegraph, 671.

Apparent, equality, 275; sequence of events, 409.

Approximation, theory of, 456; to exact laws, 462; mathematical principles of, 471; arithmetic of, 481.

Aqueous vapour, 500.

Aquinas, on disjunctive propositions, 69.

Arago, photometer, 288; rotating disc, 535; his philosophic character, 592.

Archimedes, *De Arenæ Numero*, 195; centre of gravity, 363.

Arcual unit, 306, 330.

Argyll, Duke of, 741.

Aristarchus on sun’s and moon’s distances, 294.

Aristotelian doctrines, 666.

Aristotle, dictum, 21; singular terms, 39; overlooked simple identities, 40; order of premises, 114; logical error, 117; definition of time, 307; on science, 595; on white swans, 666.

Arithmetic, reasoning in, 167; of approximate quantities, 481.

Arithmetical triangle, 93, 143, 182, 202, 378, 383; diagram of, 184; connection with Logical Alphabet, 189; in probability, 208.

Asteroids, discovery of, 412, 748.

Astronomy, physical, 459.

Atmospheric tides, 553.

Atomic theory, 662.

Atomic weights, 563.

Atoms, size of, 195; impossibility of observing, 406.

Augustin on time, 307.

Average, 359, 360; divergence from, 188; etymology of, 363.

Axes of crystals, 686.

Axioms of algebra, 164.

Babbage, Charles, calculating machine, 107, 231, 743; lighthouse signals, 194; natural constants, 329; Mosaic history, 412; universal and general truths, 646; change of law, 230; persistence of effects, 757.

Bacon, Francis Lord, *Novum Organum*, 107; on induction, 121; biliteral cipher, 193; First Aphorism, 219; on causes, 221; Copernican system, 249, 638; deficient powers of senses, 278; observation, 402; Natural History, 403; use of hypothesis, 506; his method, 507; *experimentum crucis*, 519; error of his method, 576; ostensive, clandestine instances, &c., 608, 610; *latens precessus*, 619.

Bacon, Roger, on the rainbow, 526, 598.

Baily, Francis, 272; density of earth, 342, 566; experiments with torsion balance, 370, 397, 432, 567–8; motions of stars, 572.

Bain, Alexander, on powers of mind, 4; Mill’s reform of logic, 227.

Baker’s poem, *The Universe*, 621.

Balance, use of the chemical, 292, 351, 354, 369; delicacy of, 304; vibrations of, 369.

Ballot, Buys, experiment on sound, 541.

Ballot-box, simile of, 150, 251–6, 765.

Barbara, 55, 57, 88, 105, 141.

Baroko, 85.

Barometer, 659; Gay Lussac’s standard, 346; variations, 337, 346, 349.

Bartholinus on double refraction, 585.

Base-line, measurement of, 304.

Bauhusius, verses of, 175.

Baxendell, Joseph, 552.

Beneke, on substitution, 21.

Bennet, momentum of light, 435.

Bentham, George, 15; bifurcate classification, 695; infima species, 702; works on classification, 703; analytical key to flora, 712.

Bentham, Jeremy, on analogy, 629; bifurcate classification, 703.

Benzenberg’s experiment, 388.

Bernoulli, Daniel, planetary orbits, 250; resisting media and projectiles, 467; vibrations, 476.

Bernoulli, James, 154; numbers of, 124; Protean verses, 175; *De Arte Conjectandi* quoted, 176, 183; on figurate numbers, 183; theorem of, 209; false solution in probability, 213; solution of inverse problem, 261.

Bessel, F. W., 375; law of error, 384; formula for periodic variations, 488; use of hypothesis, 506; solar parallax, 560–2; ellipticity of earth, 565; pendulum experiments, 604.

Bias, 393, 402.

Biela’s comet, 746.

Bifurcate classification, 694.

Binomial theorem, 190; discovery of, 231.

Biot, on tension of vapour, 500.

Blind experiments, 433.

Bode’s law, 147, 257, 660.

Boethius, quoted, 33; on kinds of mean, 360.

Boiling point, 442, 659.

Bonnet’s theory of reproduction, 621.

Boole, George, on sign of equality, 15; his calculus of logic, 23, 113, 634; on logical terms, 33; law of commutativeness, 35; use of *some*, 41–2; disjunctive propositions, 70; Venn on his method, 90; *Laws of Thought*, 155; statistical conditions, 168; propositions numerically definite, 172; on probability, 199; general method in probabilities, 206; Laplace’s solution of inverse problem, 256; law of error, 377.

Borda, his repeating circle, 290.

Boscovich’s hypothesis, 512.

Botany, 666, 678, 681; modes of classification, 678; systematic, 722; nomenclature of, 727.

Bowen, Prof. Francis, on inference, 118; classification, 674.

Boyle’s, Robert, law of gaseous pressure, 468, 470, 619; on hypothesis, 510; barometer, 659.

Bradley, his observations, 384; accuracy of, 271; aberration of light, 535.

Bravais, on law of error, 375.

Brewer, W. H., 142.

Brewster, Sir David, iridescent colours, 419; spectrum, 429; Newton’s theory of colours, 518; refractive indices, 10, 527; optic axes, 446.

British Museum, catalogue of, 717.

Brodie, Sir B. C., on errors of experiment, 388, 464; ozone, 663.

Brown, Thomas, on cause, 224.

Buckle, Thomas, on constancy of average, 656; science of history, 760.

Buffon, on probability, 215; definition of genius, 576.

Bunsen, Robert, spectrum, 244; photometrical researches, 273, 324, 441; calorimeter, 343.

Butler, Bishop, on probability, 197.

Calorescence, 664.

Camestres, 84.

Canton, on compressibility of water, 338.

Carbon, 640, 728; conductibility of, 442.

Cardan, on inclined plane, 501.

Cards, combinations of, 190.

Carlini, pendulum experiments, 567.

Carnot’s law, 606.

Carpenter, Dr. W. B., 412.

Catalogues, art of making, 714.

Cauchy, undulatory theory, 468.

Cause, 220; definition of, 224.

Cavendish’s experiment, 272, 566.

Cayley, Professor, 145; on mathematical tables, 331; numbers of chemical compounds, 544.

Celarent, 55.

Centre of gravity, 363, 524; of oscillation, gyration, &c., 364.

Centrobaric bodies, 364.

Certainty, 235, 266.

Cesare, 85.

Chalmers, on collocations, 740.

Chance, 198.

Character, human, 733.

Characteristics, 708.

Chauvenet, Professor W., on treatment of observations, 391.

Chemical affinity, 614; analysis, 713.

Chladni, 446.

Chloroform, discovery of, 531.

Chronoscope, 616.

Cipher, 32; Bacon’s, 193.

Circle, circumference of, 389.

Circumstances, indifferent, 419.

Circumstantial evidence, 264.

Clairaut, 650, 651; on gravity, 463.

Classes, 25; problem of common part of three, 170.

Classification, 673; involving induction, 675; multiplicity of modes, 677; natural and artificial systems, 679; in crystallography, 685; symbolic statement of, 692; bifurcate, 694; an inverse and tentative operation, 689; diagnostic, 710; by indexes, 714; of books, 715; in biological sciences, 718; genealogical, 719; by types, 722; limits of, 730.

Clifford, Professor, on types of compound statements, 143, 529; first and last catastrophe, 744; mathematical function, 768.

Clocks, astronomical, 340, 353.

Clouds, 447; cirrous, 411.

Coincidences, 128; fortuitous, 261; measurement by, 292; method of, 291.

Collective terms, 29, 39.

Collocations of matter, 740.

Colours, iridescent, 419; natural, 518; perception of, 437; of spectrum, 584.

Combinations, 135, 142; doctrine of, 173; of letters of alphabet, 174; calculations of, 180; higher orders of, 194.

Combinatorial analysis, 176.

Comets, 449; number of, 408; hyperbolic, 407; classification of, 684; conflict with, 746–7; Halley’s comet, 537; Lexell’s comet, 651.

Commutativeness, law of, 35, 72, 177.

Comparative use of instruments, 299.

Compass, variations of, 281.

Complementary statements, 144.

Compossible alternatives, 69.

Compound statements, 144; events, 204.

Compounds, chemical, 192.

Comte, Auguste, on probability, 200, 214; on prevision, 536; his positive philosophy, 752, 760, 768.

Concrete number, 159.

Conditions, of logical symbols, 32; removal of usual, 426; interference of unsuspected, 428; maintenance of similar, 443; approximation to natural, 465.

Condorcet, 2; his problem, 253.

Confusion of elements, 237.

Conical refraction, 653.

Conjunction of planets, 293, 657.

Consequent, definition of, 225.

Conservation of energy, 738.

Constant numbers of nature, 328; mathematical, 330; physical, 331; astronomical, 332; terrestrial, 333; organic, 333; social, 334.

Continuity, law of, 615, 729; sense of, 493; detection of, 610; failure of, 619.

Continuous quantity, 274, 485.

Contradiction, law of, 31, 74.

Contrapositive, proposition, 84, 136; conversion, 83.

Conversion of propositions, 46, 118.

Copernican theory, 522, 625, 638, 647.

Copula, 16.

Cornu, velocity of light, 561.

Corpuscular theory, 520, 538, 667.

Correction, method of, 346.

Correlation, 678, 681.

Cotes, Roger, use of mean, 359; method of least squares, 377.

Coulomb, 272.

Couple, mechanical, 653.

Creation, problem of, 740.

Crookes’ radiometer, 435.

Cross divisions, 144.

Crystallography, 648, 654, 658, 678, 754; systems of, 133; classification in, 685.

Crystals, 602; Dana’s classification of, 711; pseudomorphic, 658.

Curves, use of, 392, 491, 496; of various degrees, 473.

Cuvier, on experiment, 423; on inferences, 682.

Cyanite, 609.

Cycloid, 633.

Cycloidal pendulum, 461.

Cypher, 124.

D’Alembert, blunders in probability, 213, 214; on gravity, 463.

Dalton, laws of, 464, 471; atomic theory, 662.

Darapti, 59.

Darii, 56.

Darwin, Charles, his works, 131; negative results of observation, 413; arguments against his theory, 437; cultivated plants, 531; his influence, 575; classification, 718; constancy of character in classification, 720–1; on definition, 726; restoration of limbs, 730; tendency of his theory, 762, 764.

Davy, Sir H., on new instruments, 270; nature of heat, 343, 417; detection of salt in electrolysis, 428.

Day, sidereal, 310; length of, 289.

Decandolle, on classification, 696.

Decyphering, 124.

Deduction, 11, 49.

Deductive reasoning, 534; miscellaneous forms of, 60; probable, 209.

Definition, 39, 62, 711, 723; purpose of, 54; of cause and power, 224.

De Morgan, Augustus, negative terms, 14; Aristotle’s logic, 18; relatives, 23; logical universe, 43; complex propositions, 75; contraposition, 83; formal logic quoted, 101; error of his system, 117; anagram of his name, 128; numerically definite reasoning, 168–172; probability, 198; belief, 199; experiments in probability, 207; probable deductive arguments, 209–210; trisection of angle, 233; probability of inference, 259; arcual unit, 306; mathematical tables, 331; personal error, 348; average, 363; his works on probability, 394–395; apparent sequence, 409; sub-equality, 480; rule of approximation, 481; negative areas, 529; generalisation, 600; double algebra, 634; bibliography, 716; catalogues, 716; extensions of algebra, 758.

Density, unit of, 316; of earth, 387; negative, 642.

Descartes, vortices, 517; geometry, 632.

Description, 62.

Design, 762–763.

Determinants, inference by, 50.

Development, logical, 89, 97.

Diagnosis, 708.

Dichotomy, 703.

Difference, 44; law of, 5; sign of, 17; representation of, 45; inference with, 52, 166; form of, 158.

Differences of numbers, 185.

Differential calculus, 477.

Differential thermometer, 345.

Diffraction of light, 420.

Dimensions, theory of, 325.

Dip-needle, observation of, 355.

Direct deduction, 49.

Direction of motion, 47.

Discontinuity, 620.

Discordance, of theory and experiment, 558; of theories, 587.

Discoveries, accidental, 529; predicted, 536; scope for, 752.

Discrimination, 24; power of, 4.

Disjunctive, terms, 66; conjunction, 67; propositions, 66; syllogism, 77; argument, 106.

Dissipation of energy, 310.

Distance of statements, 144.

Divergence from average, 188.

Diversity, 156.

Divine interference, 765.

Dollond, achromatic lenses, 608.

Donkin, Professor, 375; on probability, 199, 216; principle of inverse method, 244.

Double refraction, 426.

Dove’s law of winds, 534.

Draper’s law, 606.

Drobitsch, 15.

Duality, 73, 81; law of, 5, 45, 92, 97.

Dulong and Petit, 341, 471.

Duration, 308.

ε, 330, 769.

Earth, density of, 387; ellipticity, 565.

Eclipses, 656; Egyptian records of, 246; of Jupiter’s satellites, 294, 372; solar, 486.

Electric, sense, 405; acid, 428; fluid, 523.

Electric telegraph, anticipations of, 671.

Electricity, theories of, 522; duality of, 590.

Electrolysis, 428, 530.

Electro-magnet, use of, 423.

Elements, confusion of, 237; definition, 427; classification, 676, 677, 690.

Elimination, 58.

Ellicott, observation on clocks, 455.

Ellipsis, 41; of terms, 57.

Elliptic variation, 474.

Ellipticity of earth, 565.

Ellis, A. J., contributions to formal logic, 172.

Ellie, Leslie, 23, 375.

Ellis, W., on moon’s influence, 410.

Emanation, law of, 463.

Emotions, 732.

Empirical, knowledge, 505, 525–526; measurement, 552.

Encke, on mean, 386, 389; his comet, 570, 605; on resisting medium, 523; solar parallax, 562.

Energy, unit of, 322; conservation of, 465; reconcentration of, 751.

English language, words in, 175.

Eözoon canadense, 412, 668.

Equality, sign of, 14; axiom, 163; four meanings of, 479.

Equations, 46, 53, 160; solution of, 123.

Equilibrium, unstable, 276, 654.

Equisetaceæ, 721.

Equivalence of propositions, 115, 120, 132; remarkable case of, 529, 657.

Eratosthenes, sieve of, 82, 123, 139; measurement of degree, 293.

Error, function, 330, 376, 381; elimination of, 339, 353; personal, 347; law of, 374; origin of law, 383; verification of law, 383; probable, 386; mean, 387; constant, 396; variation of small errors, 479.

Ether, luminiferous, 512, 514, 605.

Euclid, axioms, 51, 163; indirect proof, 84; 10th book, 117th proposition, 275; on analogy, 631.

Euler, on certainty of inference, 238; corpuscular theory, 435; gravity, 463; on ether, 514.

Everett, Professor, unit of angle, 306; metric system, 328.

Evolution, theory of, 761.

Exact science, 456.

Exceptions, 132, 644, 728; classification of, 645; imaginary, 647; apparent, 649; singular, 652; divergent, 655; accidental, 658; novel, 661; limiting, 663; real, 666; unclassed, 668.

Excluded middle, law of, 6.

Exclusive alternatives, 68.

Exhaustive investigation, 418.

Expansion, of bodies, 478; of liquids, 488.

Experiment, 400, 416; in probability, 208; test or blind, 433; negative results of, 434; limits of, 437; collective, 445; simplification of, 422; failure in simplification, 424.

Experimentalist, character of, 574, 592.

Experimentum crucis, 518, 667.

Explanation, 532.

Extent of meaning, 26; of terms, 48.

Extrapolation, 495.

Factorials, 179.

Facts, importance of false, 414; conformity with, 516.

Fallacies, 62; analysed by indirect method, 102; of observation, 408.

Faraday, Michael, measurement of gold-leaf, 296; on gravity, 342, 589; magnetism of gases, 352; vibrating plate, 419; electric poles, 421; circularly polarised light, 424, 588, 630; freezing mixtures, 427; magnetic experiments, 431, 434; lines of magnetic force, 446, 580; errors of experiment, 465; electrolysis, 502; velocity of light, 520; prediction, 543; relations of physical forces, 547; character of, 578, 587; ray vibrations, 579; mathematical power, 580; philosophic reservation of opinion, 592; use of heavy glass, 609; electricity, 612; radiant matter, 642; hydrogen, 691.

Fatality, belief in, 264.

Ferio, 56.

Figurate numbers, 183, 186.

Figure of earth, 459, 565.

Fizeau, use of Newton’s rings, 297, 582; fixity of properties, 313; velocity of light, 441, 561.

Flamsteed, use of wells, 294; standard stars, 301; parallax of pole-star, 338; selection of observations, 358; astronomical instruments, 391; solar eclipses, 486.

Fluorescence, 664.

Fontenelle on the senses, 405.

Forbes, J. D., 248.

Force, unit of, 322, 326; emanating, 464; representation of, 633.

Formulæ, empirical, 487; rational, 489.

Fortia, *Traité des Progressions*, 183.

Fortuitous coincidences, 261.

Fossils, 661.

Foster, G. C., on classification, 691.

Foucault, rotating mirror, 299; pendulum, 342, 431, 522; on velocity of light, 441, 521, 561.

Fourier, Joseph, theory of dimensions, 325; theory of heat, 469, 744.

Fowler, Thomas, on method of difference, 439; reasoning from case to case, 227.

Frankland, Professor Edward, on spectrum of gases, 606.

Franklin’s experiments on heat, 424.

Fraunhofer, dark lines of spectrum, 429.

Freezing-point, 546.

Freezing mixtures, 546.

Fresnel, inflexion of light, 420; corpuscular theory, 521; on use of hypothesis, 538; double refraction, 539.

Friction, 417; determination of, 347.

Function, definitions of, 489.

Functions, discovery of, 496.

Galileo, 626; on cycloid, 232, 235; differential method of observation, 344; projectiles, 447, 466; use of telescope, 522; gravity, 604; principle of continuity, 617.

Gallon, definition of, 318.

Galton, Francis, divergence from mean, 188; works by, 188, 655; on hereditary genius, 385, 655.

Galvanometer, 351.

Ganières, de, 182.

Gases, 613; properties of, 601, 602; perfect, 470; liquefiable, 665.

Gauss, pendulum experiments, 316; law of error, 375–6; detection of error, 396; on gravity, 463.

Gay Lussac, on boiling point, 659; law of, 669.

Genealogical classification, 680, 719.

General, terms, 29; truths, 647; notions, 673.

Generalisation, 2, 594, 704; mathematical, 168; two meanings of, 597; value of, 599; hasty, 623.

Genius, nature of, 575.

Genus, 433, 698; generalissimum, 701; natural, 724.

Geology, 667; records in, 408; slowness of changes, 438; exceptions in, 660.

Geometric mean, 361.

Geometric reasoning, 458; certainty of, 267.

Giffard’s injector, 536.

Gilbert, on rotation of earth, 249; magnetism of silver, 431; experimentation, 443.

Gladstone, J. H., 445.

Glaisher, J. W. L., on mathematical tables, 331; law of error, 375, 395.

Gold, discovery of, 413.

Gold-assay process, 434.

Gold-leaf, thickness of, 296.

Graham, Professor Thomas, on chemical affinity, 614; continuity, 616; nature of hydrogen, 691.

Grammar, 39; rules of, 31.

Grammatical, change, 119; equivalence, 120.

Gramme, 317.

Graphical method, 492.

Gravesande, on inflection of light, 420.

Gravity, 422, 512, 514, 604, 740; determination of, 302; elimination of, 427; law of, 458, 462, 474; inconceivability of, 510; Newton’s theory, 555; variation of, 565; discovery of law, 581; Faraday on, 589; discontinuity in, 620; Aristotle on, 649; Hooke’s experiment, 436.

Grimaldi on the spectrum, 584.

Grove, Mr. Justice, on ether, 514; electricity, 615.

Guericke, Otto von, 432.

Habit, formation of, 618.

Halley, trade-winds, 534.

Halley’s comet, 537, 570.

Hamilton, Sir William, disjunctive propositions, 69; inference, 118; free-will, 223.

Hamilton, Sir W. Rowan, on conical refraction, 540; quaternions, 634.

Harley, Rev. Robert, on Boole’s logic, 23, 155.

Harris, standards of length, 312.

Hartley, on logic, 7.

Hatchett, on alloys, 191.

Haughton, Professor, on tides, 450; muscular exertion, 490.

Haüy, on crystallography, 529.

Hayward, R. B., 142.

Heat, unit of, 324; measurement of, 349; experiments on, 444; mechanical equivalent of, 568.

Heavy glass, 588, 609.

Helmholtz, on microscopy, 406; undulations, 414; sound, 476.

Hemihedral crystals, 649.

Herschel, Sir John, on rotation of plane of polarisation of light, 129, 630; quartz crystals, 246; numerical precision, 273; photometry, 273; light of stars, 302; actinometer, 337; mean and average, 363; eclipses of Jupiter’s satellites, 372; law of error, 377; error in observations, 392; on observation, 400; moon’s influence on clouds, 410; comets, 411; spectrum analysis, 429; collective instances, 447; principle of forced vibrations, 451, 663; meteorological variations, 489; double stars, 499, 685; direct action, 502; use of theory, 508; ether, 515; *experimentum crucis*, 519; interference of light, 539; interference of sound, 540; density of earth, 567; residual phenomena, 569; helicoidal dissymmetry, 630; fluorescence, 664.

Hindenburg, on combinatorial analysis, 176.

Hipparchus, used method of repetition, 289; longitudes of stars, 294.

Hippocrates, area of lunule, 480.

History, science of, 760.

Hobbes, Thomas, definition of cause, 224; definition of time, 307; on hypothesis, 510.

Hofmann, unit called crith, 321; on prediction, 544; on anomalies, 670.

Homogeneity, law of, 159, 327.

Hooke, on gravitation, 436, 581; philosophical method, 507; on strange things, 671.

Hopkinson, John, 194; method of interpolation, 497.

Horrocks, use of mean, 358; use of hypothesis, 507.

Hume on perception, 34.

Hutton, density of earth, 566.

Huxley, Professor Thomas, 764; on hypothesis, 509; classification, 676; mammalia, 682; palæontology, 682.

Huyghens, theory of pendulum, 302; pendulum standard, 315; cycloidal pendulum, 341; differential method, 344; distant stars, 405; use of hypothesis, 508; philosophical method of, 585; on analogy, 639.

Hybrids, 727.

Hydrogen, expansion of, 471; refractive power, 527; metallic nature of, 691.

Hygrometry, 563.

Hypotheses, use of, 265, 504; substitution of simple hypotheses, 458; working hypotheses, 509; requisites of, 510; descriptive, 522, 686; representative, 524; probability of, 559.

Identical propositions, 119.

Identities, simple, 37; partial, 40; limited, 42; simple and partial, 111; inference from, 51, 55.

Identity, law of, 5, 6, 74; expression of, 14; propagating power, 20; reciprocal, 46.

Illicit process, of major term, 65, 103; of minor term, 65.

Immediate inference, 50, 61.

Imperfect induction, 146, 149.

Inclusion, relation of, 40.

Incommensurable quantities, 275.

Incompossible events, 205.

Independence of small effects, 475.

Independent events, 204.

Indestructibility of matter, 465.

Indexes, classification by, 714; formation of, 717.

India-rubber, properties of, 545.

Indirect method of deduction, 49, 81; illustrations of, 98; fallacies analysed by, 102; the test of equivalence, 115.

Induction, 11, 121; symbolic statement of, 131; perfect, 146; imperfect, 149; philosophy of, 218; grounds of, 228; illustrations of, 229; quantitative, 483; problem of two classes, 134; problem of three classes, 137.

Inductive truths, classes of, 219.

Inequalities, reasoning by, 47, 163, 165–166.

Inference, 9; general formula of, 17; immediate, 50; with two simple identities, 51; from simple and partial identity, 53; with partial identities, 55; by sum of predicates, 61; by disjunctive propositions, 76; indirect method of, 81; nature of, 118; principle of mathematical, 162; certainty of, 236.

Infima species, 701, 702.

Infiniteness of universe, 738.

Inflection of light, 420.

Instantiæ, citantes, evocantes, radii, curriculi, 270; monodicæ, irregulares, heteroclitæ, 608; clandestinæ, 610.

Instruments of measurement, 284.

Insufficient enumeration, 176.

Integration, 123.

Intellect, etymology of, 5.

Intension of logical terms, 26, 48; of propositions, 47.

Interchangeable system, 20.

Interpolation, 495; in meteorology, 497.

Inverse, process, 12; operation, 122, 689; problem of two classes, 134; problem of three classes, 137; problem of probability, 240, 251; rules of inverse method, 257; simple illustrations, 253; general solution, 255.

Iodine, the substance X, 523.

Iron, properties of, 528, 670.

*Is*, ambiguity of verb, 16, 41.

Isomorphism, 662.

Ivory, 375.

James, Sir H., on density of earth, 567.

Jenkin, Professor Fleming, 328.

Jevons, W. S., on use of mean, 361; on pedesis or molecular movement of microscopic particles, 406, 549; cirrous clouds, 411; spectrum analysis, 429; elevated rain-gauges, 430; experiments on clouds, 447; on muscular exertion, 490; resisting medium, 570; anticipations of the electric telegraph, 671.

Jones, Dr. Bence, Life of Faraday, 578.

Jordanus, on the mean, 360.

Joule, 545; on thermopile, 299, 300; mechanical equivalent of heat, 325, 347, 568; temperature of air, 343; rarefaction, 444; on Thomson’s prediction, 543; molecular theory of gases, 548; friction, 549; thermal phenomena of fluids, 557.

Jupiter, satellites of, 372, 458, 638, 656; long inequality of, 455; figure of, 556.

Kames, Lord, on bifurcate classification, 697.

Kant, disjunctive propositions, 69; analogy, 597; doctrine of space, 769.

Kater’s pendulum, 316.

Keill, law of emanating forces, 464; axiom of simplicity, 625.

Kepler, on star-discs, 390; comets, 408; laws of, 456; refraction, 501; character of, 578.

Kinds of things, 718.

King Charles and the Royal Society, 647.

Kirchhoff, on lines of spectrum, 245.

Kohlrausch, rules of approximate calculation, 479.

Lagrange, formula for interpolation, 497; accidental discovery, 531; union of algebra and geometry, 633.

Lambert, 15.

Lamont, 452.

Language, 8, 628, 643.

Laplace, on probability, 200, 216; principles of inverse method, 242; solution of inverse problem, 256; planetary motions, 249, 250; conjunctions of planets, 293; observation of tides, 372; atmospheric tides, 367; law of errors, 378; dark stars, 404; hyperbolic comets, 407; his works on probability, 395; velocity of gravity, 435; stability of planetary system, 448, 746; form of Jupiter, 556; corpuscular theory, 521; ellipticity of earth, 565; velocity of sound, 571; analogy, 597; law of gravity, 615; inhabitants of planets, 640; laws of motion, 706; power of science, 739.

Lavoisier, mistaken inference of, 238; pyrometer, 287; on experiments, 423; prediction of, 544; theory, 611; on acids, 667

Law, 3; of simplicity, 33, 72, 161; commutativeness, 35, 160; disjunctive relation, 71; unity, 72, 157, 162; identity, 74; contradiction, 74, 82; duality, 73, 74, 81, 97, 169; homogeneity, 159; error, 374; continuity, 615; of Boyle, 619; natural, 737.

Laws, of thought, 6; empirical mathematical, 487; of motion, 617; of botanical nomenclature, 727; natural hierarchy of, 742.

Least squares, method of, 386, 393.

Legendre, on geometry, 275; rejection of observations, 391; method of least squares, 377.

Leibnitz, 154, 163; on substitution, 21; propositions, 42; blunder in probability, 213; on Newton, 515; continuity, 618.

Leslie, differential thermometer, 345; radiating power, 425; on affectation of accuracy, 482.

Letters, combinations of, 193.

Leverrier, on solar parallax, 562.

Lewis, Sir G. C., on time, 307.

Life is change, 173.

Light, intensity of, 296; unit, 324; velocity, 535, 560, 561; science of, 538; total reflection, 650; waves of, 637; classification of, 731.

Lighthouses, Babbage on, 194.

Limited identities, 42; inference of 59.

Lindsay, Prof. T. M., 6, 21.

Linear variation, 474.

Linnæus on synopsis, 712; genera and species, 725.

Liquid state, 601, 614.

Locke, John, on induction, 121; origin of number, 157; on probability, 215; the word power, 221.

Lockyer, J. Norman, classification of elements, 676.

Logarithms, 148; errors in tables, 242.

Logic, etymology of name, 5.

Logical abacus, 104.

Logical alphabet, 93, 116, 173, 417, 701; table of, 94; connection with arithmetical triangle, 189; in probability, 205.

Logical conditions, numerical meaning of, 171.

Logical machine, 107.

Logical relations, number of, 142.

Logical slate, 95.

Logical truths, certainty of, 153.

Lottery, the infinite, 2.

Lovering, Prof., on ether, 606.

Lubbock and Drinkwater-Bethune, 386, 395.

Lucretius, rain of atoms, 223, 741; indestructibility of matter, 622.

Machine, logical, 107.

Macleay, system of classification, 719.

Magnetism of gases, 352.

Mallet, on earthquakes, 314.

Malus, polarised light, 530.

Mammalia, characters of, 681.

Manchester Literary and Philosophical Society, papers quoted, 137, 143, 168.

Mansel, on disjunctive propositions, 69.

Mars, white spots of, 596.

Maskelyne, on personal error, 347; deviation of plumbline, 369; density of earth, 566.

Mass, unit of, 317, 325.

Mathematical science, 767; incompleteness of, 754.

Matter, uniform properties of, 603; variable properties, 606.

Matthiessen, 528.

Maximum points, 371.

Maxwell, Professor Clerk, on the balance, 304; natural system of standards, 311, 319; velocity of electricity, 442; on Faraday, 580; his book on *Matter and Motion*, 634.

Mayer, proposed repeating circle, 290; on mechanical equivalent of heat, 568, 572.

Mean, etymology of, 359–360; geometric, 362; fictitious, 363; precise, 365; probable, 385; rejection of, 389; method of, 357, 554.

Mean error, 387.

Meaning, of names, 25; of propositions, 47.

Measurement, of phenomena, 270; methods of, 282; instruments, 284; indirect, 296; accuracy of, 303; units and standards of, 305; explained results of, 554; agreement of modes of, 564.

Mediate statements, 144.

Melodies, possible number of, 191.

Melvill, Thomas, on the spectrum, 429.

*Membra dividentia*, 68.

Metals, probable character of new, 258; transparency, 548; classification, 675; density, 706.

Method, indirect, 98; of avoidance of error, 340; differential, 344; correction, 346; compensation, 350; reversal, 354; means, 357; least squares, 377, 386, 393; variations, 439; graphical, 492; Baconian, 507.

Meteoric streams, 372.

Meteoric cycle, 537.

Metre, 349; error of, 314.

Metric system, 318, 323.

Michell, speculations, 212; on double stars, 247; Pleiades, 248; torsion balance, 566.

Middle term undistributed, 64.

Mill, John Stuart, disjunctive propositions, 69; induction, 121, 594; music, 191; probability, 200–201, 222; supposed reform of logic, 227; deductive method, 265, 508; elimination of chance, 385; joint method of agreement and difference, 425; method of variations, 484; on collocations, 740; erroneous tendency of his philosophy, 752.

Miller, Prof. W. H., kilogram, 318.

Mind, powers of, 4; phenomena of, 672.

Minerals, classification of, 678.

Minor term, illicit process of, 65.

Mistakes, 7.

*Modus, tolendo ponens*, 77; *ponendo tollens*, 78.

Molecular movement, or pedesis, 406.

Molecules, number of, 195.

Momentum, 322, 326.

Monro, C. J., correction by, 172; on Comte, 753.

Monstrous productions, 657.

Moon, supposed influence on clouds, 410; atmosphere of, 434; motions, 485; fall towards earth, 555.

Morse alphabet, 193.

Mother of pearl, 419.

Müller, Max, on etymology of intellect, 5.

Multiplication in logic, 161.

Murphy, J. J., on disjunctive relation, 71.

Murray, introduced use of ice, 343.

Muscular susurrus, 298.

Music, possible combinations of, 191.

Names, 25; of persons, ships, &c., 680.

Nature, 1; laws of, 737; uniformity of, 745.

Nebular theory, 427.

Negation, 44.

Negative arguments, 621.

Negative density, 642.

Negative premises, 63, 103.

Negative propositions, 43.

Negative results of experiment, 434.

Negative terms, 14, 45, 54, 74.

Neil on use of hypothesis, 509.

Neptune, discovery of, 537, 660.

Newton, Sir Isaac, binomial theorem, 231; spectrum, 262, 418, 420, 424, 583; rings of, 288, 470; velocity of sound, 295; wave-lengths, 297; use of pendulum, 303; on time, 308; definition of matter, 316; pendulum experiment, 348, 443, 604; centrobaric bodies, 365; on weight, 422; achromatic lenses, 432; resistance of space, 435; absorption of light, 445; planetary motions, 249, 457, 463, 466, 467; infinitesimal calculus, 477; as an alchemist, 505; his knowledge of Bacon’s works, 507; *hypotheses non fingo*, 515; on vortices, 517; theory of colours, 518; corpuscular theory of light, 520; fits of easy reflection, &c., 523; combustible substances, 527; gravity, 555, 650; density of earth, 566; velocity of sound, 571; third law of motion, 622; his rules of philosophising, 625; fluxions, 633; theory of sound, 636; negative density, 642; rays of light having sides, 662.

Newtonian Method, 581.

Nicholson, discovery of electrolysis, 530.

*Ninth Bridgewater Treatise* quoted, 743, 757.

Nipher, Professor, on muscular exertion, 490.

Noble, Captain, chronoscope, 308, 616.

Nomenclature, laws of botanical, 727.

Non-observation, arguments from, 411.

Norwood’s measurement of a degree, 272.

Nothing, 32.

Number, nature of, 153, 156; concrete and abstract, 159, 305.

Numbers, prime, 123; of Bernoulli, 124; figurate, 183; triangular, &c., 185.

Numerical abstraction, 158.

Observation, 399; mental conditions, 402; instrumental and sensual conditions, 404; external conditions, 407.

Obverse statements, 144.

Ocean, depth of, 297.

Odours, 732.

Oersted, on electro-magnetism, 530, 535.

*Or*, meaning of, 70.

Order, of premises, 114; of terms, 33.

Orders of combinations, 194.

Original research, 574.

Oscillation, centre of, 364.

Ostensive instances, 608.

Ozone, 663.

π, value of, 234, 529.

Pack of cards, arrangement of, 241.

Paley on design, 762, 763.

Parallax, of stars, 344; of sun, 560.

Parallel forces, 652.

Paralogism, 62.

Parity of reasoning, 268.

Partial identities, 40, 55, 57, 111; induction of, 130.

Particular quantity, 56.

Particulars, reasoning from, 227.

Partition, 29.

Pascal, 176; arithmetical machine, 107; arithmetical triangle, 182; binomial formula, 182; error in probabilities, 213; barometer, 519.

Passive state of steel, 659.

Pedesis, or molecular movement of microscopic particles, 406, 612.

Peirce, Professor, 23; on rejection of observations, 391.

Pendulum, 290, 302, 315; faults of, 311; vibrations, 453, 454; cycloidal, 461.

Perfect induction, 146, 149.

Perigon, 306.

Permutations, 173, 178; distinction from combinations, 177.

Personal error, 347.

Photometry, 288.

Physiology, exceptions in, 666.

Planets, conjunctions of, 181, 187, 657; discovery of, 412; motions, 457; perturbations of, 657; classification, 683; system of, 748.

Plants, classification of, 678.

Plateau’s experiments, 427.

Plato on science, 595.

Plattes, Gabriel, 434, 438.

Pliny on tides, 451.

Plumb-line, divergence of, 461.

Plurality, 29, 156.

Poinsot, on probability, 214.

Poisson, on principle of the inverse method, 244; work on Probability, 395; Newton’s rings, 470; simile of ballot box, 524.

Polarisation, 653; discovery of, 530.

Pole-star, 652; observations of, 366.

Poles, of magnets, 365; of battery, 421.

Political economy, 760.

Porphyry, on the Predicables, 698; tree of, 702.

Port Royal logic, 22.

Positive philosophy, 760, 768.

Pouillet’s pyrheliometer, 337.

Powell, Baden, 623; on planetary motions, 660.

Power, definition of, 224.

Predicables, 698.

Prediction, 536, 739; in science of light, 538; theory of undulations, 540; other sciences, 542; by inversion of cause and effect, 545.

Premises, order of, 114.

Prime numbers, 123, 139; formula for, 230.

*Principia*, Newton’s, 581, 583.

Principle, of probability, 200; inverse method, 242; forced vibrations, 451; approximation, 471; co-existence of small vibrations, 476; superposition of small effects, 476.

Probable error, 555.

Probability, etymology of, 197; theory of, 197; principles, 200; calculations, 203; difficulties of theory, 213; application of theory, 215; in induction, 219; in judicial proceedings, 216; works on, 394; results of law, 656.

Problems, to be worked by reader, 126; inverse problem of two classes, 135; of three classes, 137.

Proclus, commentaries of, 232.

Proctor, R. A., star-drifts, 248.

Projectiles, theory of, 466.

Proper names, 27.

Properties, generality of, 600; uniform, 603; extreme instances, 607; correlation, 681.

Property, logical, 699; peculiar, 699.

Proportion, simple, 501.

Propositions, 36; negative, 43; conversion of, 46; twofold meaning, 47; disjunctive, 66; equivalence of, 115; identical, 119; tautologous, 119.

Protean verses, 175.

Protoplasm, 524, 764.

Prout’s law, 263, 464.

Provisional units, 323.

Proximate statements, 144.

Pyramidal numbers, 185.

Pythagoras, on duality, 95; on the number seven, 262, 624.

Quadric variation, 474.

Qualitative, reasoning, 48; propositions, 119.

Quantification of predicate, 41.

Quantitative, reasoning, 48; propositions, 119; questions, 278; induction, 483.

Quantities, continuous, 274; incommensurable, 275.

Quaternions, 160, 634.

Quetelet, 188; experiment on probability, 208; on mean and average, 363; law of error, 378, 380; verification of law of error, 385.

Radian, 306.

Radiant matter, 642.

Radiation of heat, 430.

Radiometer, 435.

Rainbow, theory of, 526, 533.

Rainfall, variation of, 430.

Ramean tree, 703, 704.

Ramsden’s balance, 304.

Rankine, on specific heat of air, 557; reconcentration of energy, 751.

Rational formulæ, 489.

Rayleigh, Lord, on graphical method, 495.

Reasoning, arithmetical, 167; numerically definite, 168; geometrical, 458.

Recorde, Robert, 15.

Reduction, of syllogisms, 85; *ad absurdum*, 415; of observations, 552, 572.

Reflection, total, 650.

Refraction, atmospheric, 340, 356, 500; law of, 501; conical, 540; double, 585.

Regnault, dilatation of mercury, 342; measurement of heat, 350; exact experiment, 397; on Boyle’s law, 468, 471; latent heat of steam, 487; graphical method, 494; specific heat of air, 557.

Reid, on bifurcate classification, 697.

Reign of law, 741, 759.

Rejection of observations, 390.

Relation, sign of, 17; logic of, 22; logical, 35; axiom of, 164.

Repetition, method of, 287, 288.

Representative hypotheses, 524.

Reproduction, modes of, 730.

Reservation of judgment, 592.

Residual effects, 558; phenomena, 560, 569.

Resisting medium, 310, 523, 570.

Resonance, 453.

Reusch, on substitution, 21.

Reversal, method of, 354.

Revolution, quantity of, 306.

Robertson, Prof. Croom, 27, 101.

Robison, electric curves, 446.

Rock-salt, 609.

Rœmer, divided circle, 355; velocity of light, 535.

Roscoe, Prof., photometrical researches, 273; solubility of salts, 280; constant flame, 441; absorption of gases, 499; vanadium, 528; atomic weight of vanadium, 392, 649.

Rousseau on geometry, 233.

Rules, of inference, 9, 17; indirect method of inference, 89; for calculation of combinations, 180; of probabilities, 203; of inverse method, 257; for elimination of error, 353.

Rumford, Count, experiments on heat, 343, 350, 467.

Ruminants, Cuvier on, 683.

Russell, Scott, on sound, 541.

Sample, use of, 9.

Sandeman, on perigon, 306; approximate arithmetic, 481.

Saturn, motions of satellites, 293; rings, 293.

Schehallien, attraction of, 369, 566.

Schottus, on combinations, 179.

Schwabe, on sun-spots, 452.

Science, nature of, 1, 673.

Selenium, 663, 670.

Self-contradiction, 32.

Senior’s definition of wealth, 75.

Senses, fallacious indications of, 276.

Seven, coincidences of number, 262; fallacies of, 624.

Sextus, fatality of name, 264.

Sieve of Eratosthenes, 82, 123, 139.

Similars, substitution of, 17.

Simple identity, 37, 111; inference of, 58; contrapositive, 86; induction of, 127.

Simple statement, 143.

Simplicity, law of, 33, 58, 72.

Simpson, discovery of property of chloroform, 531.

Simultaneity of knowledge, 34.

Singular names, 27; terms, 129.

Siren, 10, 298, 421.

Slate, the logical, 95.

Smeaton’s experiments, on water-wheels, 347; windmills, 401, 441.

Smee, Alfred, logical machines, 107.

Smell, delicacy of, 437.

Smithsonian Institution, 329.

Smyth, Prof. Piazzi, 452.

Socrates, on the sun, 611.

Solids, 602.

Solubility of salts, 279.

*Some*, the adjective, 41, 56.

Sorites, 60.

Sound, observations on, 356; undulations, 405, 421; velocity of, 571; classification of sounds, 732.

Space, relations of, 220.

Species, 698; infima, 701; natural, 724.

Specific gravities, 301; heat of air, 557.

Spence, on boiling point, 546.

Spencer, Herbert, nature of logic, 4, 7; sign of equality, 15; rhythmical motion, 448; abstraction, 705; philosophy of, 718, 761, 762.

Spectroscope, 437.

Spectrum, 583.

Spiritualism, 671.

Spontaneous generation, 432.

Standards of measurement, 305; the bar, 312; terrestrial, 314; pendulum, 315; provisional, 318; natural system, 319.

Stars, discs of, 277; motions of, 280, 474; variations of, 281; approach or recess, 298; standard stars, 301; apparent diameter, 390; variable, 450; proper motions, 572; Bruno on, 639; new, 644; pole-star, 652; conflict with wandering stars, 748.

Stas, M., his balance, 304; on atomic weights, 464.

Statements, kinds of, 144.

Statistical conditions, 168.

Stevinus, on inclined plane, 622.

Stewart, Professor Balfour, on resisting medium, 570; theory of exchanges, 571.

Stifels, arithmetical triangle, 182.

Stokes, Professor, on resistance, 475; fluorescence, 664.

Stone, E. J., heat of the stars, 370; temperature of earth’s surface, 452; transit of Venus, 562.

Struve on double stars, 247.

Substantial terms, 28.

Substantives, 14.

Substitution of similars, 17, 45, 49, 104, 106; anticipations of, 21.

Substitutive weighing, 345.

*Sui generis*, 629, 728.

Sulphur, 670.

Summum genus, 93, 701.

Sun, distance, 560; variations of spots, 452.

Superposition, of small effects, 450; small motions, 476.

Swan, W., on sodium light, 430.

Syllogism, 140; moods of, 55, 84, 85, 88, 105, 141; numerically definite, 168.

Symbols, use of, 13, 31, 32; of quantity, 33.

Synthesis, 122; of terms, 30.

Table-turning, 671.

Tacit knowledge, 43.

Tacquet on combinations, 179.

Tait, P. G., 375; theory of comets, 571.

Talbot on the spectrum, 429.

Tartaglia on projectiles, 466.

Tastes, classification of, 732.

Tautologous propositions, 119.

Teeth, use in classification, 710.

Temperature, variations of, 453.

Tension of aqueous vapour, 500.

Terms, 24; abstract, 27; substantial, 28; collective, 29; synthesis of, 30; negative, 45.

Terrot, Bishop, on probability, 212.

Test experiments, 347, 433.

Tetractys, 95.

Thales, predicted eclipse, 537.

Theory, results of, 534; facts known by, 547; quantitative, 551; of exchanges, 571; freedom of forming, 577; of evolution, 761.

Thermometer, differential, 345; reading of, 390; change of zero, 390.

Thermopile, 300.

Thomas, arithmetical machine, 107.

Thomson, Archbishop, 50, 61.

Thomson, James, prediction by, 542; on gaseous state, 654.

Thomson, Sir W., lighthouse signals, 194; size of atoms, 195; tides, 450; capillary attraction, 614; magnetism, 665; dissipation of energy, 744.

Thomson and Tait, chronometry, 311; standards of length, 315; the crowbar, 460; polarised light, 653.

Thomson, Sir Wyville, 412.

Thunder-cloud, 612.

Tides, 366, 450, 476, 541; velocity of, 298; gauge, 368; atmospheric, 367, 553.

Time, 220; definition of, 307.

Todhunter, Isaac, *History of the Theory of Probability*, 256, 375, 395; on insoluble problems, 757.

Tooke, Horne, on cause, 226.

Torricelli, cycloid, 235; his theorem, 605; on barometer, 666.

Torsion balance, 272, 287.

Transit of Venus, 294, 348, 562.

Transit-circle, 355.

Tree of Porphyry, 702; of Ramus, 703.

Triangle, arithmetical, 93, 182.

Triangular numbers, 185.

Trigonometrical survey, 301; calculations of, 756.

Trisection of angles, 414.

Tuning-fork, 541.

Tycho Brahe, 271; on star discs, 277; obliquity of earth’s axis, 289; circumpolar stars, 366; Sirius, 390.

Tyndall, Professor, on natural constants, 328; magnetism of gases, 352; precaution in experiments, 431; use of imagination, 509; on Faraday, 547; magnetism, 549, 607; scope for discovery, 753.

Types, of logical conditions, 140, 144; of statements, 145; classification by, 722.

Ueberweg’s logic, 6.

Ultimate statements, 144.

Undistributed, attribute, 40; middle term, 64, 103.

Undulations, of light, 558; analogy in theory of, 635.

Undulatory theory, 468, 520, 538, 540; inconceivability of, 510.

Unique objects, 728.

Unit, definition of, 157; groups, 167; of measurement, 305; arcual, 306; of time, 307; space, 312; density, 316; mass, 317; subsidiary, 320; derived, 321; provisional, 323; of heat, 325; magnetical and electrical units, 326, 327.

Unity, law of, 72.

Universe, logical, 43; infiniteness of, 738; heat-history of, 744, 749; possible states of, 749.

Uranus, anomalies of, 660.

Vacuum, Nature’s abhorrence of, 513.

Vapour densities, 548.

Variable, variant, 440, 441, 483.

Variation, linear, elliptic, &c., 474; method of, 439.

Variations, logical, 140; periodic, 447; combined, 450; integrated, 452; simple proportional, 501.

Variety, of nature, 173; of nature and art, 190; higher orders of, 192.

Velocity, unit of, 321.

Venn, Rev. John, logical problem by, 90; on Boole, 155; his work on *Logic of Chance*, 394.

Venus, 449; transits of, 294.

Verses, Protean, 175.

Vibrations, law of, 295; principle of forced, 451; co-existence of small, 476.

Vital force, 523.

Voltaire on fossils, 661.

Vortices, theory of, 513, 517.

Vulcan, supposed planet, 414.

Wallis, 124, 175.

Water, compressibility of, 338; properties of, 610.

Watt’s parallel motion, 462.

Waves, 599, 635; nature of, 468; in canals, 535; earthquake, 297.

Weak arguments, effect of, 211.

Wells, on dew, 425.

Wenzel, on neutral salts, 295.

Whately, disjunctive propositions, 69; probable arguments, 210.

Wheatstone, cipher, 124; galvanometer, 286; revolving mirror, 299, 308; kaleidophone, 445; velocity of electricity, 543.

Whewell, on tides, 371, 542; method of least squares, 386.

Whitworth, Sir Joseph, 304, 436.

Whitworth, Rev. W. A., on *Choice and Chance*, 395.

Wilbraham, on Boole, 206.

Williamson, Professor A. W., chemical unit, 321; prediction by, 544.

Wollaston, the goniometer, 287; light of moon, 302; spectrum, 429.

Wren, Sir C., on gravity, 581.

X, the substance, 523.

Yard, standard, 397.

Young, Dr. Thomas, tension of aqueous vapour, 500; use of hypotheses, 508; ethereal medium, 515.

Zero point, 368.

Zodiacal light, 276.

Zoology, 666.

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SPELLING CORRECTIONS

acording → according aklaline → alkaline an an → an aws → laws beween → between BOOK III → BOOK IV errror → error incapadle → incapable interpretion → interpretation justifed → justified longtitude → longitude Marriotte → Mariotte melecules → molecules Meropolitana → Metropolitana necesssarily → necessarily nnmber → number or → of probabilty → probability quantites → quantities secresy → secrecy sucession → succession suficiently → sufficiently telecope → telescope verifiy → verify