Book VI
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.
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 recognised605 the truth of Chalmers’ statement, without drawing the proper inferences from it. He says606 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.
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?”
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.
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
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.
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 remarks615 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.
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.
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.
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. Todhunter618 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.
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 2x + cos 2x = 1. But this is not the only trigonometry which may exist; mathematicians also recognise hyperbolic trigonometry, of which the fundamental equation is cos 2x - sin 2x = 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.
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.
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.
Footnotes
1 Since the above was written Mr. Harley has read an account of Stanhope’s logical remains at the Dublin Meeting (1878) of the British Association. The paper will be printed in Mind. (Note added November, 1878.)
2 Leibnitii Opera Philosophica quæ extant. Erdmann, Pars I. Berolini, 1840, p. 94.
3 Erdmann, p. 102.
4 Ibid. p. 98.
5 Erdmann, p. 100.
6 Fifth Edition, 1860, p. 158.
7 Section 120.
8 See his “Remarks on Boole’s Mathematical Analysis of Logic.” Report of the 36th Meeting of the British Association, Transactions of the Sections, pp. 3–6.
9 Hamilton’s Lectures, vol. iv. p. 319.
10 Ibid. p. 326.
11 Pure Logic, or the Logic of Quality apart from Quantity; with Remarks on Boole’s System, and on the Relation of Logic and Mathematics. London, 1864, p. 3.
12 La Philosophie Positive, Mai-Juin, 1877, tom. xviii. p. 456.
13 Inventum Novum Quadrati Logici, &c., Gissæ Hassorum, 1714, 8vo.
14 See Ueberweg’s System of Logic, &c., translated by Lindsay, p. 302.
15 Since the above was written M. Liard has republished this exposition as one chapter of an interesting and admirably lucid account of the progress of logical science in England. After a brief but clear introduction, treating of the views of Herschel, Mill, and others concerning Inductive Logic, M. Liard describes in succession the logical systems of George Bentham, Hamilton, De Morgan, Boole, and that contained in the present work. The title of the book is as follows:—Les Logiciens Anglais Contemporains. Par Louis Liard, Professeur de Philosophie à la Faculté des Lettres de Bordeaux. Paris: Librairie Germer Baillière. 1878. (Note added November, 1878.)
16 Spectator, September 19, 1874, p. 1178. A second portion of the review appeared in the same journal for September 26, 1874, p. 1204.
17 Mind: a Quarterly Review of Psychology and Philosophy. No. II. April 1876. Vol. I. p. 206.
18 Portions of this work have already been published in my articles, entitled “John Stuart Mill’s Philosophy Tested,” printed in the Contemporary Review for December, 1877, vol. xxxi. p. 167, and for January and April, 1878, vol. xxxi. p. 256, and vol. xxxii. p. 88. (Note added in November, 1878.)
19 Mind, vol. i. p. 222.
20 Fortnightly Review, New Series, April 1875, p. 480. Lecture reprinted by the Sunday Lecture Society, p. 24.
21 Sir W. Thomson’s words are as follows (Cambridge Mathematical Journal, Nov. 1842, vol. iii. p. 174). “When x is negative, the state represented cannot be the result of any possible distribution of temperature which has previously existed.” There is no limitation in the sentence to the laws of conduction, but, as the whole paper treats of the results of conduction in a solid, it may no doubt be understood that there is a tacit limitation. See also a second paper on the subject in the same journal for February, 1844, vol. iv. p. 67, where again there is no expressed limitation.
22 Pp. 25–26. The parentheses are in the original, and show Professor Tait’s corrections in the verbatim reports of his lectures. The subject is treated again on pp. 168–9.
23 Theory of Heat 1871, p. 245.
24 The Senses and the Intellect, Second Ed., pp. 5, 325, &c.
25 Max Müller, Lectures on the Science of Language, Second Series, vol. ii. p. 63; or Sixth Edition, vol. ii. p. 67. The view of the etymological meaning of “intellect” is given above on the authority of Professor Max Müller. It seems to be opposed to the ordinary opinion, according to which the Latin intelligere means to choose between, to see a difference between, to discriminate, instead of to unite.
26 Hartley on Man, vol. i. p. 359.
27 Principles of Psychology, Second Ed., vol. ii. p. 86.
28 Pure Logic, or the Logic of Quality apart from Quantity, 1864, pp. 10, 16, 22, 29, 36, &c.
29 Brewster, Treatise on New Philosophical Instruments, p. 273. Concerning this method see also Whewell, Philosophy of the Inductive Sciences, vol. ii. p. 355; Tomlinson, Philosophical Magazine, Fourth Series, vol. xl. p. 328; Tyndall, in Youmans’ Modern Culture, p. 16.
30 Formal Logic, p. 38.
31 Hallam’s Literature of Europe, First Ed., vol. ii. p. 444.
32 Outline of a New System of Logic, London, 1827, pp. 133, &c.
33 An Investigation of the Laws of Thought, pp. 27, &c.
34 Formal Logic, pp. 82, 106. In his later work, The Syllabus of a New System of Logic, he discontinued the use of the sign.
35 Principles of Psychology, Second Ed., vol. ii. pp. 54, 55.
36 Pure Logic, or the Logic of Quality, p. 14.
37 Pure Logic, pp. 18, 19.
38 Ueberweg’s System of Logic, transl. by Lindsay, pp. 442–446, 571, 572. The anticipations of the principle of substitution to be found in the works of Leibnitz, Reusch, and perhaps other German logicians, will be noticed in the preface to this second edition.
39 Substitution of Similars (1869), p. 9.
40 Port-Royal Logic, transl. by Spencer Baynes, pp. 212–219. Part III. chap. x. and xi.
41 Description of a Notation for the Logic of Relatives, resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic. By C. S. Peirce. Memoirs of the American Academy, vol. ix. Cambridge, U.S., 1870.
42 On the Syllogism No IV., and on the Logic of Relations. By Augustus De Morgan. Transactions of the Cambridge Philosophical Society, vol. x. part ii., 1860.
43 Observations on Boole’s Laws of Thought. By the late R. Leslie Ellis; communicated by the Rev. Robert Harley, F.R.S. Report of the British Association, 1870. Report of Sections, p. 12. Also, On Boole’s Laws of Thought. By the Rev. Robert Harley, F.R.S., ibid. p. 14.
44 Jevons’ Elementary Lessons in Logic, pp. 41–43; Pure Logic, p. 6. See also J. S. Mill, System of Logic, Book I. chap. ii. section 5, and Shedden’s Elements of Logic, London, 1864, pp. 14, &c. Professor Robertson objects (Mind, vol. i. p. 210) that I confuse singular and proper names; if so, it is because I hold that the same remarks apply to proper names, which do not seem to me to differ logically from singular names.
45 Professor Robertson has criticised my introduction of “Substantial Terms” (Mind, vol. i. p. 210), and objects, perhaps correctly, that the distinction if valid is extra-logical. I am inclined to think, however, that the doctrine of terms is, strictly speaking, for the most part extra-logical.
46 Mathematical Analysis of Logic, Cambridge, 1847, p. 17. An Investigation of the Laws of Thought, London, 1854, p. 31.
47 Pure Logic, p. 15.
48 “Velut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed uno toties prædicaverim.”
49 Book i., Part iv., Section 5.
50 Laws of Thought, p. 29. It is pointed out in the preface to this Second Edition that Leibnitz was acquainted with the Laws of Simplicity and of Commutativeness.
51 Prior Analytics, i. cap. xxvii. 3.
52 Encyclopædia Britannica, Eighth Ed. art. Logic, sect. 37, note. 8vo. reprint, p. 79.
53 De Morgan, On the Root of any Function. Cambridge Philosophical Transactions, 1867, vol. xi. p. 25.
54 Syllabus of a proposed System of Logic, §§ 122, 123.
55 Elementary Lessons in Logic, p. 86.
56 Outline of the Laws of Thought, § 87.
57 Treatise on Natural Philosophy, vol. i. p. 161.
58 Treatise on Natural Philosophy, vol. i. p. 6.
59 Todhunter’s Plane Co-ordinate Geometry, chap. ii. pp. 11–14.
60 An explanation of this and other technical terms of the old logic will be found in my Elementary Lessons in Logic, Sixth Edition, 1876; Macmillan.
61 Elementary Lessons in Logic, pp. 67, 79.
62 Pure Logic, p. 19.
63 An Outline of the Necessary Laws of Thought, Fifth Ed. p. 161.
64 Mansel’s Aldrich, p. 103, and Prolegomena Logica, p. 221.
65 Elements of Logic, Book II. chap. iv. sect. 4.
66 Aldrich, Artis Logicæ Rudimenta, p. 104.
67 Examination of Sir W. Hamilton’s Philosophy, pp. 452–454.
68 Pure Logic, pp 76, 77.
69 Pure Logic, p. 65. See also the criticism of this point by De Morgan in the Athenæum, No. 1892, 30th January, 1864; p. 155.
70 Boole’s Laws of Thought, p. 106. Jevons’ Pure Logic, p. 69.
71 On the Syllogism, No. iii. p. 12. Camb. Phil. Trans. vol. x, part i.
72 See Horsley, Philosophical Transactions, 1772; vol. lxii. p. 327. Montucla, Histoire des Mathematiques, vol. i. p. 239. Penny Cyclopædia, article “Eratosthenes.”
73 Euclid, Book x. Prop. 117.
74 Philosophical Magazine, December 1852; Fourth Series, vol. iv. p. 435, “On Indirect Demonstration.”
75 Philosophical Magazine, Dec. 1852; p. 437.
76 Mind; a Quarterly Review of Psychology and Philosophy; October, 1876, vol. i. p. 487.
77 Whewell, History of the Inductive Sciences, vol. i. p. 222.
78 Formal Logic, p. 124. As Professor Croom Robertson has pointed out to me, the second and third premises may be thrown into a single proposition, D = DeBC ꖌ DEbc.
79 Pp. 55–59, 81–86.
80 See his work called The Process of Thought adapted to Words and Language, together with a Description of the Relational and Differential Machines. Also Philosophical Transactions, [1870] vol. 160, p. 518.
81 Philosophical Transactions [1870], vol. 160, p. 497. Proceedings of the Royal Society, vol. xviii. p. 166, Jan. 20, 1870. Nature, vol, i. p. 343.
82 Syllabus of a proposed system of Logic, §§ 57, 121, &c. Formal Logic, p. 66.
83 Lectures on Metaphysics, vol. iv. p. 369.
84 Bowen, Treatise on Logic, Cambridge, U.S., 1866; p. 362.
85 The contents of this and the following section nearly correspond with those of a paper read before the Manchester Literary and Philosophical Society on December 26th, 1871. See Proceedings of the Society, vol. xi. pp. 65–68, and Memoirs, Third Series, vol. v. pp. 119–130.
86 Proceedings of the Manchester Literary and Philosophical Society, 6th February, 1877, vol. xvi., p. 113.
87 Montucla. Histoire des Mathématiques, vol. iii. p. 373.
88 British Quarterly Review, No. lxxxvii, July 1866.
89 Mind, October 1876, vol. i. p. 484.
90 Pure Logic, Appendix, p. 82, § 192.
91 Elementary Lessons in Logic (Macmillan), p. 123. It is pointed out in the preface to this Second Edition, that the views here given were partially stated by Leibnitz.
92 Syllabus of a Proposed System of Logic, p. 29.
93 It has been pointed out to me by Mr. C. J. Monroe, that section 14 (p. 339) of this paper is erroneous, and ought to be cancelled. The problem concerning the number of paupers illustrates the answer which should have been obtained. Mr. A. J. Ellis, F.R.S., had previously observed that my solution in the paper of De Morgan’s problem about “men in the house” did not answer the conditions intended by De Morgan, and I therefore give in the text a more satisfactory solution.
94 Montucla, Histoire, &c., vol. iii. p. 388.
95 Wallis, Of Combinations, &c., p. 119.
96 James Bernoulli, De Arte Conjectandi, translated by Baron Maseres. London, 1795, pp. 35, 36.
97 Arithmeticæ Theoria. Ed. Amsterd. 1704. p. 517.
98 Rees’s Cyclopædia, art. Cipher.
99 Œuvres Complètes de Pascal (1865), vol. iii. p. 302. Montucla states the name as De Gruières, Histoire des Mathématiques, vol. iii. p. 389.
100 Histoire des Mathématiques, vol. iii. p. 378.
101 Bernoulli, De Arte Conjectandi, translated by Francis Maseres. London, 1795, p. 75.
102 Wallis’s Algebra, Discourse of Combinations, &c., p. 109.
103 Œuvres Complètes, vol. iii. p. 251.
104 See also Galton’s Lecture at the Royal Institution, 27th February, 1874; Catalogue of the Special Loan Collection of Scientific Instruments, South Kensington, Nos. 48, 49; and Galton, Philosophical Magazine, January 1875.
105 Wallis, Of Combinations, p. 116, quoting Vossius.
106 Philosophical Transactions (1803), vol. xciii. p. 193.
107 Hofmann’s Introduction to Chemistry, p. 36.
108 Works, edited by Shaw, vol. i. pp. 141–145, quoted in Rees’s Encyclopædia, art. Cipher.
109 Nature, vol. i. p. 553.
110 Formal Logic, p. 172.
111 Philosophical Magazine, 4th Series, vol. i. p. 355.
112 Transactions of the Royal Society of Edinburgh, vol. xxi. part 4.
113 Philosophical Magazine, 4th Series, vol. vii. p. 465; vol. viii. p. 91.
114 Memoirs of the Manchester Literary and Philosophical Society, 3rd Series, vol. iv. p. 347.
115 Letters on the Theory of Probabilities, translated by Downes, 1849, pp. 36, 37.
116 Encyclopædia Metropolitana, art. Probabilities, p. 396.
117 Elements of Logic, Book III. sections 11 and 18.
118 Encyclopædia Metropolitana, art. Probabilities, p. 400.
119 Philosophical Transactions (1767). Abridg. vol. xii. p. 435.
120 Transactions of the Edinburgh Philosophical Society, vol. xxi. p. 375.
121 Montucla, Histoire des Mathématiques, vol. iii. p. 386.
122 Leibnitz Opera, Dutens’ Edition, vol. vi. part i. p. 217. Todhunter’s History of the Theory of Probability, p. 48. To the latter work I am indebted for many of the statements in the text.
123 Positive Philosophy, translated by Martineau, vol. ii. p. 120.
124 System of Logic, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61.
125 Montucla, Histoire, vol. iii. p. 405; Todhunter, p. 263.
126 Essay concerning Human Understanding, bk. iv. ch. 14. § 1.
127 Philosophical Magazine, 4th Series, vol. i. p. 354.
128 Essay concerning Human Understanding, bk. ii. chap. xxi.
129 De Rerum Natura, bk. ii. ll. 216–293.
130 Cambridge Philosophical Transactions (1830), vol. iii. pp. 369–372.
131 Observations on the Nature and Tendency of the Doctrine of Mr. Hume, concerning the Relation of Cause and Effect. Second ed. p. 44.
132 Ibid. p. 97.
133 System of Logic, bk. II. chap, iii.
134 Inductive Logic, pp. 13, 14.
135 Bain, Deductive Logic, pp. 208, 209.
136 System of Logic. Introduction, § 4. Fifth ed. pp. 8, 9.
137 Ibid. bk. II. chap. iii. § 5, pp. 225, &c.
138 These are the figurate numbers considered in pages 183, 187, &c.
139 Commercium Epistolicum. Epistola ad Oldenburgum, Oct. 24, 1676. Horsley’s Works of Newton, vol. iv. p. 541. See De Morgan in Penny Cyclopædia, art. “Binomial Theorem,” p. 412.
140 Bk. ii. chap. iv.
141 Philosophical Transactions (1866), vol. 146, p. 334.
142 Budget of Paradoxes, p. 257.
143 Proceedings of the Royal Society (1872–3), vol. xxi. p. 319.
144 Life of Galileo, Society for the Diffusion of Useful Knowledge, p. 102.
145 Professor Bowen has excellently stated this view. Treatise on Logic. Cambridge, U.S.A., 1866, p. 354.
146 Roscoe’s Spectrum Analysis, 1st edit., p. 98.
147 Euler’s Letters to a German Princess, translated by Hunter. 2nd ed., vol. ii. pp. 17, 18.
148 Lavoisier’s Chemistry, translated by Kerr. 3rd ed., pp. 114, 121, 123.
149 Euler’s Letters, vol. ii. p. 21.
150 Lardner, Edinburgh Review, July 1834, p. 277.
151 Mémoires par divers Savans, tom. vi.; quoted by Todhunter in his History of the Theory of Probability, p. 458.
152 Poisson, Recherches sur la Probabilité des Jugements, Paris, 1837, pp. 82, 83.
153 Kirchhoff’s Researches on the Solar Spectrum. First part, translated by Roscoe, pp. 18, 19.
154 Edinburgh Review, No. 185, vol. xcii. July 1850, p. 32; Herschel’s Essays, p. 421; Transactions of the Cambridge Philosophical Society, vol. i. p. 43.
155 Evans’ Ancient Stone Implements of Great Britain. London, 1872 (Longmans).
156 Herschel, Outlines of Astronomy, 1849, p. 565; but Todhunter, in his History of the Theory of Probability, p. 335, states that the calculations do not agree with those published by Struve.
157 Philosophical Transactions, 1767, vol. lvii. p. 431.
158 Philosophical Magazine, 3rd Series, vol. xxxvii. p. 401, December 1850; also August 1849.
159 History, &c., p. 334.
160 Essai Philosophique, p. 57.
161 Proceedings of the Royal Society; 20 January, 1870; Philosophical Magazine, 4th Series, vol. xxxix. p. 381.
162 Principia, bk. ii. General scholium.
163 Essai Philosophique, p. 55. Laplace appears to count the rings of Saturn as giving two independent movements.
164 Lubbock, Essay on Probability, p. 14. De Morgan, Encyc. Metrop. art. Probability, p. 412. Todhunter’s History of the Theory of Probability, p. 543. Concerning the objections raised to these conclusions by Boole, see the Philosophical Magazine, 4th Series, vol. ii. p. 98. Boole’s Laws of Thought, pp. 364–375.
165 Laplace, Essai Philosophique, pp. 55, 56.
166 Chambers’ Astronomy, 2nd ed. pp. 346–49.
167 Traité élémentaire du Calcul des Probabilités, 3rd ed. (1833), p. 148.
168 Laws of Thought, pp. 368–375.
169 De Morgan’s Essay on Probabilities, Cabinet Cyclopædia, p. 67.
170 Essay on Probabilities, p. 128.
171 J. S. Mill, System of Logic, 5th edition, bk. iii. chap. xviii. § 3.
172 Todhunter’s History, pp. 472, 598.
173 Todhunter’s History, pp. 378, 379.
174 Philosophical Transactions, [1763], vol. liii. p. 370, and [1764], vol. liv. p. 296. Todhunter, pp. 294–300.
175 Newton’s Opticks, Bk. I., Part ii. Prop. 3; Nature, vol. i. p. 286.
176 Aristotle’s Metaphysics, xiii. 6. 3.
177 Possunt autem omnes testes et uno annulo signare testamentum Quid enim si septem annuli una sculptura fuerint, secundum quod Pomponio visum est?—Justinian, ii. tit. x. 5.
178 See Wills on Circumstantial Evidence, p. 148.
179 Memoirs of the Royal Astronomical Society, vol. iv. p. 290, quoted by Lardner, Edinburgh Review, July 1834, p. 278.
180 Baily, British Association Catalogue of Stars, pp. 7, 23.
181 Outlines of Astronomy, 4th ed. sect. 781, p. 522. Results of Observations at the Cape of Good Hope, &c., p. 37.
182 See De Morgan, Study of Mathematics, in U.K.S. Library, p. 81.
183 Loomis, On the Aurora Borealis. Smithsonian Transactions, quoting Parry’s Third Voyage, p. 61.
184 Watts’ Dictionary of Chemistry, vol. ii. p. 790.
185 Philosophical Transactions, (1856) vol. 146, Part i. p. 297.
186 Airy, On Tides and Waves, Encyclopædia Metropolitana, p. 345. Scott Russell, British Association Report, 1837, p. 432.
187 Hugenii Cosmotheoros, pp. 117, 118. Laplace’s Système, translated, vol. i. p. 67.
188 Grant’s History of Physical Astronomy, p. 129.
189 Baily’s Account of Flamsteed, p. lix.
190 Jamin, Cours de Physique, vol. i. p. 152.
191 Faraday, Chemical Researches, p. 393.
192 Proceedings of the Royal Society, 30th November, 1866.
193 Herschel, Physical Geography, § 40.
194 Principia, bk. iii. Prop. 37, Corollaries, 2 and 3. Motte’s translation, vol. ii. p. 310.
195 Roscoe’s Spectrum Analysis, 1st ed. p. 296.
196 Philosophical Transactions (1859), vol. cxlix. p. 94.
197 Watts’ Dictionary of Chemistry, vol. ii. p. 393.
198 Philosophical Transactions (1859), vol. cxlix. p. 119, &c.
199 Baily’s Account of Flamsteed, pp. 378–380.
200 Herschel’s Astronomy, § 817, 4th. ed. p. 553.
201 Principia, bk. ii. Sect. 6. Prop. 31. Motte’s Translation, vol. ii. p. 107.
202 Ibid. bk. i. Law iii. Corollary 6. Motte’s Translation, vol. i. p. 33.
203 Thomson and Tait’s Natural Philosophy, vol. i. p. 333.
204 Philosophical Transactions, (1856), vol. cxlvi. pp. 330, 331.
205 First Annual Report of the Mint, p. 106.
206 Jevons, in Watts’ Dictionary of Chemistry, vol. i. p. 483.
207 British Association, Glasgow, 1856. Address of the President of the Mechanical Section.
208 Pelicotetics, or the Science of Quantity; an Elementary Treatise on Algebra, and its groundwork Arithmetic. By Archibald Sandeman, M. A. Cambridge (Deighton, Bell, and Co.), 1868, p. 304.
209 De Morgan’s Trigonometry and Double Algebra, p. 5.
210 English Works of Thos. Hobbes, Edit. by Molesworth, vol. i. p. 95.
211 Confessions, bk. xi. chapters 20–28.
212 Sir G. C. Lewis gives many curious particulars concerning the measurement of time in his Astronomy of the Ancients, pp. 241, &c.
213 Principia, bk. i. Scholium to Definitions. Translated by Motte, vol. i. p. 9. See also p. 11.
214 Rankine, Philosophical Magazine, Feb. 1867, vol. xxxiii. p. 91.
215 Treatise on Natural Philosophy, vol. i. p. 179.
216 Proceedings of the Manchester Philosophical Society, 28th Nov. 1871, vol. xi. p. 33.
217 The Elements of Natural Philosophy, part i. p. 119.
218 See Harris’ Essay upon Money and Coins, part. ii. [1758] p. 127.
219 Philosophical Magazine, (1868), 4th Series, vol. xxxvi. p. 32.
220 Proceedings of the Royal Society, 20th June, 1872, vol. xx. p. 438.
221 Kater’s Treatise on Mechanics, Cabinet Cyclopædia, p. 154.
222 Grant’s History of Physical Astronomy, p. 156.
223 Clerk Maxwell’s Theory of Heat, p. 79.
224 Treatise on Electricity and Magnetism, vol. i. p. 3.
225 Chemistry for Students, by A. W. Williamson. Clarendon Press Series, 2nd ed. Preface p. vi.
226 Introduction to Chemistry, p. 131.
227 Philosophical Transactions (1859), vol. cxlix. p. 884, &c.
228 Théorie Analytique de la Chaleur, Paris; 1822, §§ 157–162.
229 Tyndall’s Sound, 1st ed. p. 26.
230 British Association, Cambridge, 1833. Report, pp. 484–490.
231 Smithsonian Miscellaneous Collections, vol. xii., the Constants of Nature, part. i. Specific gravities compiled by F. W. Clarke, 8vo. Washington, 1873.
232 J. W. L. Glaisher, Philosophical Magazine, 4th Series, vol. xlii. p. 421.
233 Stokes, Philosophical Transactions (1852), vol. cxlii. p. 529.
234 Admiralty Manual of Scientific Enquiry, 2nd ed. p. 299.
235 Pouillet, Taylor’s Scientific Memoirs, vol. iv. p. 45.
236 Baily’s Account of the Rev. John Flamsteed, p. 58.
237 Jamin, Cours de Physique, vol. ii. pp. 15–28.
238 Philosophical Magazine, 1851, 4th Series, vol. ii. passim.
239 Hearn, Philosophical Transactions, 1847, vol. cxxxvii. pp. 217–221.
240 The Correlation of Physical Forces, 3rd ed. p. 159.
241 Collected Works of Sir H. Davy, vol. ii. pp. 12–14. Elements of Chemical Philosophy, p. 94.
242 Nicholson’s Journal, vol. i. p. 241; quoted in Treatise on Heat, Useful Knowledge Society, p. 24.
243 Clerk Maxwell, Theory of Heat, p. 228. Proceedings of the Manchester Philosophical Society, Nov. 26, 1867, vol. vii. p. 35.
244 Leslie, Inquiry into the Nature of Heat, p. 10.
245 Jevons, Watts’ Dictionary of Chemistry, vol. i. pp. 513–515.
246 Philosophical Transactions, vol. li. p. 100.
247 Philosophical Magazine, 3rd Series, vol. xxvi. p. 372.
248 Greenwich Observations for 1866, p. xlix.
249 Philosophical Transactions, 1856, p. 309.
250 Penny Cyclopædia, art. Transit, vol. xxv. pp. 129, 130.
251 Ibid. art. Observation, p. 390.
252 Nature, vol. i. p. 85.
253 Nature, vol. i. p 337. See references to the Memoirs describing the method.
254 Principia, Book I. Law III. Corollary VI. Scholium. Motte’s translation, vol. i. p. 33.
255 Graham’s Chemical Reports and Memoirs, Cavendish Society, pp. 247, 268, &c.
256 Regnault’s Cours Elémentaire de Chimie, 1851, vol i. p. 141.
257 Tyndall’s Faraday, pp. 114, 115.
258 See, for instance, the Compensated Sympiesometer, Philosophical Magazine, 4th Series, vol. xxxix. p. 371.
259 Grant, History of Physical Astronomy, pp. 146, 147.
260 Quetelet, Sur la Physique du Globe, p. 174. Jamin, Cours de Physique, vol. i. p. 504.
261 Baily’s Account of Flamsteed, p. 376.
262 The Transit of Venus across the Sun, by Horrocks, London, 1859, p. 146.
263 De Morgan, Supplement to the Penny Cyclopædia, art. Old Appellations of Numbers.
264 Penny Cyclopædia, art. Mean.
265 Jevons, Journal of the Statistical Society, June 1865, vol. xxviii, p. 296.
266 Letters on the Theory of Probabilities, transl. by Downes, Part ii.
267 Herschel’s Essays, &c. pp. 404, 405.
268 On the Theory of Errors of Observations, Cambridge Philosophical Transactions, vol. x. Part ii. 416.
269 Thomson and Tait, Treatise on Natural Philosophy, vol. i. p. 394.
270 Essai Philosophique sur les Probabilités, pp. 49, 50.
271 Grant, History of Physical Astronomy, p. 163.
272 Gauss, Taylor’s Scientific Memoirs, vol. ii. p. 43, &c.
273 Proceedings of the Royal Society, vol. xviii. p. 159 (Jan. 13, 1870). Philosophical Magazine (4th Series), vol. xxxix. p. 376.
274 Airy On Tides and Waves, Encycl. Metrop. pp. 364*-366*.
275 Outlines of Astronomy, 4th edition, § 538.
276 Philosophical Magazine, 3rd Series, vol. xxxvii. p. 324.
277 Letters on the Theory of Probabilities, by Quetelet, translated by O. G. Downes, Notes to Letter XXVI. pp. 286–295.
278 On the Law of Facility of Errors of Observations, and on the Method of Least Squares, Memoirs of the Royal Astronomical Society, vol. xxxix. p. 75.
279 Méthode des Moindres Carrés. Mémoires sur la Combinaison des Observations, par Ch. Fr. Gauss. Traduit en Français par J. Bertrand, Paris, 1855, pp. 6, 133, &c.
280 De Morgan, Penny Cyclopædia, art. Least Squares.
281 Edinburgh Review, July 1850, vol. xcii. p. 17. Reprinted Essays, p. 399. This method of demonstration is discussed by Boole, Transactions of Royal Society of Edinburgh, vol. xxi. pp. 627–630.
282 Letters on the Theory of Probabilities, Letter XV. and Appendix, note pp. 256–266.
283 Encke, On the Method of Least Squares, Taylor’s Scientific Memoirs, vol. ii. pp. 338, 339.
284 Quetelet, Letters on the Theory of Probabilities, translated by Downes, Letter XIX. p. 88. See also Galton’s Hereditary Genius, p. 379.
285 System of Logic, bk. iii. chap. 17, § 3. 5th ed. vol. ii. p. 56.
286 Philosophy of the Inductive Sciences, 2nd ed. vol. ii. pp. 408, 409.
287 Essay on Probability, Useful Knowledge Society, 1833, p. 41.
288 Taylor’s Scientific Memoirs, vol. ii. p. 333.
289 Philosophical Transactions, 1873, p. 83.
290 Taylor’s Scientific Memoirs, vol. ii. pp. 330, 347, &c.
291 Quetelet, Letters, &c. p. 116.
292 Baily, Account of Flamsteed, p. 56.
293 Gould’s Astronomical Journal, Cambridge, Mass., vol. ii. p. 161.
294 Philadelphia (London, Trübner) 1863. Appendix, vol. ii. p. 558.
295 Bakerian Lecture, Philosophical Transactions (1868), vol. clviii. p. 6.
296 Results of Observations at the Cape of Good Hope, p. 283.
297 The Logic of Chance, an Essay on the Foundations and Province of the Theory of Probability, with especial reference to its Logical Bearings and its Application to Moral and Social Science. (Macmillan), 1876.
298 Gauss, translated by Bertrand, p. 25.
299 Jamin, Cours de Physique, vol. ii. p. 60.
300 Preliminary Discourse on the Study of Natural Philosophy, p. 77.
301 Lavoisier’s Elements of Chemistry, translated by Kerr, 3rd ed. p. 148.
302 Babbage, Economy of Manufactures, p. 194.
303 System of the World, translated by Harte, vol. ii. p. 335.
304 This curious phenomenon, which I propose to call pedesis, or the pedetic movement, from πηδόω, to jump, is carefully described in my paper published in the Quarterly Journal of Science for April, 1878, vol. viii. (N.S.) p. 167. See also Proceedings of the Literary and Philosophical Society of Manchester, 25th January, 1870, vol. ix. p. 78, Nature, 22nd August, 1878, vol. xviii. p. 440, or the Quarterly Journal of Science, vol. viii. (N.S.) p. 514.
305 Maxwell, Theory of Heat, p. 301.
306 Laplace, Essai Philosophique, p. 59. Todhunter’s History, pp. 491–494.
307 Chambers’ Astronomy, 1st ed. p. 203.
308 Essay on Probabilities, Cabinet Cyclopædia, p. 121.
309 Philosophical Magazine, 4th Series (1867), vol. xxxiv. p. 64.
310 See Notes to Measures of Double Stars, 1204, 1336, 1477, 1686, 1786, 1816, 1835, 1929, 2081, 2186, pp. 265, &c. See also Herschel’s Familiar Lectures on Scientific Subjects, p. 147, and Outlines of Astronomy, 7th ed. p. 285.
311 Jevons, On the Cirrous Form of Cloud, Philosophical Magazine, July, 1857, 4th Series, vol. xiv. p. 22.
312 Astronomy, 4th ed. p. 358.
313 Babbage, Ninth Bridgewater Treatise, p. 67.
314 Cuvier, Essay on the Theory of the Earth, translation, p. 61, &c.
315 Murchison’s Siluria, 1st ed. p. 432.
316 Darwin’s Fertilisation of Orchids, p. 48.
317 Peacock, Algebre, vol. ii. p. 344.
318 Ibid, p. 359. Serret, Algèbre Supérieure, 2nd ed. p. 304.
319 Treatise on Optics, by Brewster, Cab. Cyclo. p. 117.
320 Opticks, 3rd. ed. p. 25.
321 Experimental Researches in Electricity, vol. i. pp. 133, 134.
322 Ibid. vol i. pp. 127, 162, &c.
323 Principia, bk. iii. Prop. vi. Corollary i.
324 Lavoisier’s Chemistry, translated by Kerr, p. 103.
325 Cuvier’s Animal Kingdom, introduction, pp. 1, 2.
326 Experimental Researches in Electricity, vol. iii. p. 4.
327 Philosophical Magazine, 4th Series, vol. ix. p. 327.
328 Inquiry into the Nature of Heat, p. 95.
329 Herschel, Preliminary Discourse, p. 161.
330 System of Logic, bk. iii. chap. viii. § 4, 5th ed. vol. i. p. 433.
331 Essayes of Natural Experiments made in the Accademia del Cimento. Englished by Richard Waller, 1684, p. 40, &c.
332 Plateau, Taylor’s Scientific Memoirs, vol. iv. pp. 16–43.
333 Philosophical Transactions [1826], vol. cxvi. pp. 388, 389. Works of Sir Humphry Davy, vol. v. pp. 1–12.
334 National Review, July, 1861, p. 13.
335 His published works are contained in The Edinburgh Physical and Literary Essays, vol. ii. p. 34; Philosophical Transactions [1753], vol. xlviii. p. 261; see also Morgan’s Papers in Philosophical Transactions [1785], vol. lxxv. p. 190.
336 Edinburgh Journal of Science, vol. v. p. 79.
337 Encyclopædia Metropolitana, art. Light, § 524; Herschel’s Familiar Lectures, p. 266.
338 Talbot, Philosophical Magazine, 3rd Series, vol. ix. p. 1 (1836); Brewster, Transactions of the Royal Society of Edinburgh [1823], vol. ix. pp. 433, 455; Swan, ibid. [1856] vol. xxi. p. 411; Philosophical Magazine, 4th Series, vol. xx. p. 173 [Sept. 1860]; Roscoe, Spectrum Analysis, Lecture III.
339 Balfour Stewart, Elementary Treatise on Heat, p. 192.
340 British Association, Liverpool, 1870. Report on Rainfall, p. 176.
341 Philosophical Magazine., Dec. 1861. 4th Series, vol. xxii. p. 421.
342 Experimental Researches in Electricity, vol. iii. p. 84, &c.
343 Lectures on Heat, p. 21.
344 Baily, Memoirs of the Royal Astronomical Society, vol. xiv. pp. 29, 30.
345 Grant, History of Physical Astronomy, p. 531.
346 Philosophical Transactions, abridged by Lowthorp, 4th edition, vol. i. p. 202.
347 Jevons in Watts’ Dictionary of Chemistry, vol. ii. pp. 936, 937.
348 Discovery of Subterraneal Treasure. London, 1639, p. 48.
349 Laplace, System of the World, translated by Harte, vol. ii. p. 322.
350 Principia, bk. ii. sect. 6, Prop. xxxi. Motte’s translation, vol. ii. p. 108.
351 Essayes of Natural Experiments, &c. p. 117.
352 Hooke’s Posthumous Works, p. 182.
353 Principia, bk. iii. Prop. vii. Corollary 1.
354 Keill’s Introduction to Natural Philosophy, 3rd ed., London, 1733, pp. 48–54.
355 Discovery of Subterraneal Treasure, 1639, p. 52.
356 Elements of Inductive Logic, 1st edit. p. 175.
357 Philosophical Transactions, vol. li. p. 138; abridgment, vol. xi. p. 355.
358 See Bunsen and Roscoe’s researches, in Philosophical Transactions (1859), vol. cxlix. p. 880, &c., where they describe a constant flame of carbon monoxide gas.
359 Humboldt’s Cosmos (Bohn), vol. i. p. 7.
360 Gilbert, De Magnete, p. 109.
361 Principia, bk. iii. Prop. vi.
362 Philosophical Magazine, 3rd Series, vol. xxvi. p. 375.
363 Opticks, 3rd edit. p. 159.
364 Watts, Dictionary of Chemistry, vol. iii. p. 637.
365 Faraday’s Life, by Bence Jones, vol. ii. p. 5.
366 Preliminary Discourse, &c., p. 185.
367 Philosophical Magazine, July, 1857, 4th Series, vol. xiv. p. 24.
368 First Principles, 3rd edit. chap. x. p. 253.
369 Laplace, System of the World, vol. i. pp. 50, 54, &c.
370 Herschel’s Outlines of Astronomy, 4th edit. pp. 555–557.
371 Humboldt’s Cosmos (Bohn), vol. iii. p. 229.
372 Encyclopædia Metropolitana, art. Sound, § 323; Outlines of Astronomy, 4th edit., § 650. pp. 410, 487–88; Meteorology, Encyclopædia Britannica, Reprint, p. 197.
373 Philosophical Transactions, (1739), vol. xli. p. 126.
374 Principia, bk. iii. Prop. 15.
375 Lockyer’s Lessons in Elementary Astronomy, p. 301.
376 Treatise on Natural Philosophy, vol. i. pp. 337, &c.
377 An Introduction to Natural Philosophy, 3rd edit. 1733, p. 5.
378 Watts, Dictionary of Chemistry, vol. i. p. 455.
379 Philosophical Transactions, (1866) vol. clvi. p. 809.
380 Experimental Researches in Electricity, vol. i. p. 246.
381 Hutton’s Mathematical Dictionary, vol. ii. pp. 287–292.
382 Principia, bk. iii. Prop. 13.
383 Jamin, Cours de Physique, vol. i. pp. 282, 283.
384 Lloyd’s Lectures on the Wave Theory, pp. 22, 23.
385 Tait’s Thermodynamics, p. 10.
386 Lloyd’s Lectures on the Wave Theory, pp. 82, 83.
387 Jamin, Cours de Physique, vol. i. pp. 283–288.
388 Joule and Thomson, Philosophical Transactions, 1854, vol. cxliv. p. 337.
389 The properties of a perfect gas have been described by Rankine, Transactions of the Royal Society of Edinburgh, vol. xxv. p. 561.
390 Thomson and Tait’s Natural Philosophy, vol. i. p. 60.
391 Challis, Notes on the Principles of Pure and Applied Calculation, 1869, p. 83.
392 An Introduction to Physical Measurements, translated by Waller and Procter, 1873, p. 10.
393 Cambridge Philosophical Transactions (1865), vol. xi. Part I.
394 Sandeman, Pelicotetics, p. 214.
395 The Science and Art of Arithmetic for the Use of Schools. (Whitaker and Co.)
396 Principles of Approximate Calculations, by J. J. Skinner, C.E. (New York, Henry Holt), 1876.
397 Leslie, Inquiry into the Nature of Heat, p. 505.
398 System of Logic, bk. iii. chap. viii § 6.
399 Laplace, System of the World, translated by Harte, vol. ii. p. 366.
400 Chemical Reports and Memoirs, Cavendish Society, p. 294.
401 Jamin, Cours de Physique, vol. ii. p. 38.
402 On Tides and Waves, Encyclopædia Metropolitana, p. 366*.
403 Encyclopædia Britannica, art. Meteorology. Reprint, §§ 152–156.
404 Lagrange, Leçons sur le Calcul des Fonctions, 1806, p. 4.
405 Haughton, Principles of Animal Mechanics, 1873, pp. 444–450. Jevons, Nature, 30th of June, 1870, vol. ii. p. 158. See also the experiments of Professor Nipher, of Washington University, St. Louis, in American Journal of Science, vol. ix. p. 130, vol. x. p. 1; Nature, vol. xi. pp. 256, 276.
406 Jamin, Cours de Physique, vol. ii. p. 50.
407 Philosophical Transactions, 1826, p. 544.
408 Jamin, Cours de Physique, vol. ii. p. 24, &c.
409 J. W. Strutt, On a correction sometimes required in curves professing to represent the connexion between two physical magnitudes. Philosophical Magazine, 4th Series, vol. xlii. p. 441.
410 Herschel: Lacroix’ Differential Calculus, p. 551.
411 Cours complet de Météorologie, Note A, p. 449.
412 On the Calculation of Empirical Formulæ. The Messenger of Mathematics, New Series, No. 17, 1872.
413 Watts’ Dictionary of Chemistry, vol. ii. p. 790.
414 Quarterly Journal of the Chemical Society, vol. viii. p. 15.
415 Results of Observations at the Cape of Good Hope, p. 293.
416 Jamin, Cours de Physique, vol. ii. p. 138.
417 Preliminary Discourse, &c., p. 152.
418 Tyndall, On Cometary Theory, Philosophical Magazine, April 1869. 4th Series, vol. xxxvii. p. 243.
419 See Philosophical Transactions, abridged by Lowthorp. 4th edit. vol. i. p. 130. I find that opinions similar to those in the text have been briefly expressed by De Morgan in his remarkable preface to From Matter to Spirit, by C.D., pp. xxi. xxii.
420 Horrocks, Opera Posthuma (1673), p. 276.
421 Young’s Works, vol. i. p. 593.
422 Boyle’s Physical Examen, p. 84.
423 Young’s Works, vol. i. p. 415.
424 Familiar Lectures on Scientific Subjects, p. 282.
425 Young’s Works, vol. i. p. 417.
426 Principia, bk. iii. Prop. 43. General Scholium.
427 Ibid. bk. ii. Sect. ix. Prop. 53.
428 Brewster’s Life of Newton, 1st edit. chap. vii.
429 Discourse on the Study of Natural Philosophy, p. 151.
430 Ibid. p. 229.
431 Novum Organum, bk. ii. Aphorism 36.
432 Principia, bk. i. Sect. xiv. Prop. 96. Scholium. Opticks, Prop. vi. 3rd edit. p. 70.
433 Airy’s Mathematical Tracts, 3rd edit. pp. 286–288.
434 Jamin, Cours de Physique, vol. iii. p. 372.
435 Young’s Lectures on Natural Philosophy (1845), vol. i. p. 361.
436 Paris, Life of Davy, p. 274.
437 Opus Majus. Edit. 1733. Cap. x. p. 460.
438 Newton’s Opticks. Third edit. p. 249.
439 Brewster. Treatise on New Philosophical Instruments, p. 266, &c.
440 Roscoe, Bakerian Lecture, Philosophical Transactions (1868), vol. clviii. p. 6.
441 Life of Faraday, vol. ii. p. 104.
442 Watts, Dictionary of Chemistry, vol. ii, p. 39, &c.
443 De Morgan’s Budget of Paradoxes, p. 291.
444 Life of Faraday, vol. ii p. 396.
445 Experimental Researches in Electricity, 1st Series, pp. 24–44.
446 Airy, On Tides and Waves, Encyclopædia Metropolitana, p. 348*
447 Lib. i. cap. 74.
448 Taylor’s Scientific Memoirs, vol. v. p. 241.
449 Airy’s Mathematical Tracts, 3rd edit. p. 312.
450 Young’s Works, vol. i. p. 412.
451 Lloyd’s Wave Theory, Part ii. pp. 52–58. Babbage, Ninth Bridgewater Treatise, p. 104, quoting Lloyd, Transactions of the Royal Irish Academy, vol. xvii. Clifton, Quarterly Journal of Pure and Applied Mathematics, January 1860.
452 Encyclopædia Metropolitana, art. Sound, p. 753.
453 Tyndall’s Sound, pp. 261, 273.
454 Whewell’s History of the Inductive Sciences, vol. ii. p. 471. Herschel’s Physical Geography, § 77.
455 Maxwell’s Theory of Heat, p. 174. Philosophical Magazine, August 1850. Third Series, vol. xxxvii. p. 123.
456 Philosophical Transactions, 1858, vol. cxlviii. p. 127.
457 Tyndall’s Faraday, pp. 73, 74; Life of Faraday, vol. ii. pp. 82, 83.
458 Tait’s Thermodynamics, p. 77.
459 On the Analytical Forms called Trees, with Application to the Theory of Chemical Combinations. Report of the British Association, 1875, p. 257.
460 Hofmann’s Introduction to Chemistry, pp. 224, 225.
461 Philosophical Transactions (1855), vol. cxlv. pp. 100, &c.
462 Proceedings of the Manchester Philosophical Society, Feb. 1870.
463 Balfour Stewart, Elementary Treatise on Heat, 1st edit. p. 198.
464 Jevons, Proceedings of the Manchester Literary and Philosophical Society, 25th January, 1870, vol. ix. p. 78.
465 Philosophical Transactions, vol. cxlvi. p. 249.
466 Grant’s History of Physical Astronomy, p. 162.
467 Philosophical Transactions (1854), vol. cxliv. p. 364.
468 Monthly Notices of the Royal Astronomical Society, vol. xxviii. p. 264.
469 It would seem to be absurd to repeat the profuse expenditure of 1874 at the approaching transit in 1882. The aggregate sum spent in 1874 by various governments and individuals can hardly be less than £200,000, a sum which, wisely expended on scientific investigations, would give a hundred important results.
470 Philosophical Transactions (1856), vol. cxlvi. p. 342.
471 Monthly Notices of the Royal Astronomical Society, for 8th Nov. 1844, No. X. vol. vi. p. 89.
472 Philosophical Magazine, 2nd Series, vol. xxvi. p. 61.
473 Clausius in Philosophical Magazine, 4th Series, vol. ii. p. 119.
474 Watts’ Dictionary of Chemistry, vol. iii. p. 129.
475 Preliminary Discourse, §§ 158, 174. Outlines of Astronomy, 4th edit. § 856.
476 Proceedings of the Manchester Literary and Philosophical Society, 28th November, 1871, vol. xi. p. 33. Since the above remarks were written, Professor Balfour Stewart has pointed out to me his paper in the Proceedings of the Manchester Literary and Philosophical Society for 15th November, 1870 (vol. x. p. 32), in which he shows that a body moving in an enclosure of uniform temperature would probably experience resistance independently of the presence of a ponderable medium, such as gas, between the moving body and the enclosure. The proof is founded on the theory of the dissipation of energy, and this view is said to be accepted by Professors Thomson and Tait. The enclosure is used in this case by Professor Stewart simply as a means of obtaining a proof, just as it was used by him on a previous occasion to obtain a proof of certain consequences of the Theory of Exchanges. He is of opinion that in both of these cases when once the proof has been obtained, the enclosure may be dispensed with. We know, for instance, that the relation between the inductive and absorptive powers of bodies—although this relation may have been proved by means of an enclosure, does not depend upon its presence, and Professor Stewart thinks that in like manner two bodies, or at least two bodies possessing heat such as the sun and the earth in motion relative to each other, will have the differential motion retarded until perhaps it is ultimately destroyed.
477 British Association Catalogue of Stars, p. 49.
478 Experimental Researches in Chemistry and Physics, p. 372. Philosophical Magazine, 3rd Series, May 1846, vol. xxviii. p. 350.
479 See also Nature, September 18, 1873; vol. viii. p. 398.
480 Theory of Political Economy, pp. 3–14.
481 Principia, bk. i. Prop. iv.
482 Opticks, bk. i. part ii. Prop. 3. 3rd ed. p. 115.
483 Experimental Inquiry into the Nature of Heat. Preface, p. xv.
484 Bence Jones, Life of Faraday, vol. i. p. 362.
485 Ibid. vol. ii. p. 199.
486 See also his more formal statement in the Experimental Researches in Electricity, 24th Series, § 2702, vol. iii. p. 161.
487 Printed in Modern Culture, edited by Youmans, p. 219.
488 Life of Faraday, vol. i. p. 225.
489 Aristotle’s Rhetoric, Liber I. 2. 11.
490 Essai Philosophique sur les Probabilités, p. 86.
491 Kant’s Logik, § 84, Königsberg, 1800, p. 207.
492 Syllabus of a Proposed System of Logic, p. 34.
493 Principia, bk. iii. Prop. VI. Motte’s translation, vol. ii. p. 220.
494 Professor Lovering has pointed out how obscure and uncertain the ideas of scientific men about this ether are, in his interesting Presidential Address before the American Association at Hartford, 1874. Silliman’s Journal, October 1874, p. 297. Philosophical Magazine, vol. xlviii. p. 493.
495 Novum Organum, bk. ii. Aphorisms, 24, 25.
496 Ibid. Aph. 28.
497 Philosophical Transactions (1856) vol. cxlvi. p. 246.
498 Philosophical Magazine, 4th Series, January 1870, vol. xxxix. p. 2.
499 Novum Organum, bk. ii. Aphorism 25.
500 Faraday’s Experimental Researches in Chemistry and Physics, p. 93.
501 Memorabilia, iv. 7.
502 Experimental Researches in Electricity, Series xii. vol. i. p. 420.
503 Life of Faraday, vol. ii. p. 7.
504 Nature, vol. ii. p. 278.
505 Journal of the Chemical Society, vol. viii. p. 51.
506 Correlation of Physical Forces, 3rd edit. p. 184.
507 Philosophical Magazine, 4th Series, vol. xlii. p. 451.
508 Grove, Correlation of Physical Forces, 3rd edit. p. 118.
509 Ibid. pp. 166, 199, &c.
510 Philosophical Transactions, 1861. Chemical and Physical Researches, p. 598.
511 Life of Sir W. Hamilton, p. 439.
512 Powell’s History of Natural Philosophy, p. 201. Novum Organum, bk. ii. Aphorisms 5–7.
513 Thomson and Tait, Treatise on Natural Philosophy, vol. i. pp. 346–351.
514 Philosophical Transactions (1740), vol. xli. p. 454.
515 Principia, bk. i. Law iii. Corollary 6.
516 Helmholtz, Taylor’s Scientific Memoirs (1853), vol. vi. p. 118.
517 Lucretius, bk. i. lines 232–264.
518 Novum Organum, bk. 1 Aphorism 104.
519 The Unity of Worlds and of Nature, 2nd edit. p. 116.
520 Principia, bk. iii, ad initium.
521 Keill, Introduction to Natural Philosophy, p. 89.
522 Jeremiæ Horroccii Opera Posthuma (1673), pp. 26, 27.
523 Young’s Works, vol. ii. p. 564.
524 Essay on Logic, Works, vol. viii. p. 276.
525 Life of Faraday, by Bence Jones, vol. ii. p. 206.
526 Lacroix, Traité Élémentaire de Calcul Différentiel et de Calcul Intégral, 5me édit. p. 699.
527 Histoire des Mathématiques, vol. i. p. 298.
528 See Goodwin, Cambridge Philosophical Transactions (1845), vol. viii. p. 269. O’Brien, “On Symbolical Statics,” Philosophical Magazine, 4th Series, vol. i. pp. 491, &c. See also Professor Clerk Maxwell’s delightful Manual of Elementary Science, called Matter and Motion, published by the Society for Promoting Christian Knowledge. In this admirable little work some of the most advanced results of mechanical and physical science are explained according to the method of quaternions, but with hardly any use of algebraic symbols.
529 Birch, History of the Royal Society, vol. iii. p. 262, quoted by Young, Works, vol. i. p. 246.
530 Opticks, Query 28, 3rd edit. p. 337.
531 Rankine, Philosophical Transactions (1856), vol. cxlvi. p. 282.
532 Cosmotheoros (1699), p. 16.
533 Laplace, System of the World, vol. ii. p. 316.
534 Cosmotheoros (1699), p. 17.
535 Ibid. p. 36.
536 System of the World, vol. ii. p. 326. Essai Philosophique, p. 87.
537 Principia, bk. ii. Section ii. Prop. x.
538 De Morgan, Cambridge Philosophical Transactions, vol. xi. Part ii. p. 246.
539 Life of Faraday, vol. i. p. 216.
540 Babbage, The Exposition of 1851, p. 1.
541 Daubeny’s Atomic Theory, p. 76.
542 Bakerian Lecture, Philosophical Transactions (1868), vol. clviii. p. 2.
543 Principia, bk. ii. Prop. 20. Corollaries, 5 and 6.
544 Treatise on Natural Philosophy, vol. i. p. 50.
545 Maxwell’s Theory of Heat, (1871), p. 175.
546 Galton, on the Height and Weight of Boys. Journal of the Anthropological Institute, 1875, p. 174.
547 Grant’s History of Physical Astronomy, p. 116.
548 Discourse to the Royal Society, 28th May, 1684.
549 Robert Hooke’s Posthumous Works, p. 365.
550 Experimental Researches in Electricity, vol. ii. pp. 240–245.
551 Murchison’s Silurian System, vol. ii. p. 733, &c.
552 Philosophical Transactions (1872), vol. clxii. No. 23.
553 Philosophical Transactions (1852), vol. cxlii. pp. 465, 548, &c.
554 Philosophical Magazine, 4th Series, vol. i. p. 182.
555 Maxwell, Theory of Heat, p. 123.
556 Prior Analytics, ii. 2, 8, and elsewhere.
557 Hofmann’s Introduction to Chemistry, p. 198.
558 Stewart’s Elementary Treatise on Heat, p. 80.
559 Jevons, Proceedings of the Manchester Literary and Philosophical Society, 6th March, 1877, vol. xvi. p. 164. See also Mr. W. E. A. Axon’s note on the same subject, ibid. p. 166.
560 A Treatise on Logic, or, the Laws of Pure Thought, by Francis Bowen, Professor of Moral Philosophy in Harvard College, Cambridge, United States, 1866, p. 315.
561 Proceedings of the Royal Society, November, 1873, vol. xxi. p. 512.
562 Lectures on the Elements of Comparative Anatomy, 1864, p. 1.
563 Essai sur la Philosophie des Sciences, p. 9.
564 Lectures on the Elements of Comparative Anatomy, and on the Classification of Animals, 1864, p. 3.
565 Ossemens Fossiles, 4th edit. vol. i. p. 164. Quoted by Huxley, Lectures, &c., p. 5.
566 Chambers, Descriptive Astronomy, 1st edit. p. 23.
567 Philosophical Magazine, 4th Series, vol. xxxix. p. 396; vol. xl. p. 183; vol. xli. p. 44. See also Proctor, Popular Science Review, October 1874, p. 350.
568 Humboldt, Cosmos (Bohn), vol. iii. p. 224.
569 Baily, British Association Catalogue, p. 48.
570 Outlines of Astronomy, § 850, 4th edit. p. 578.
571 Life of Faraday, vol. ii. p. 87.
572 Proceedings of the Royal Society, vol. xvii. p. 212. Chemical and Physical Researches, reprint, by Young and Angus Smith, p. 290.
573 Essai sur la Nomenclature et la Classification, Paris, 1823, pp. 107, 108.
574 George Bentham, Outline of a New System of Logic, p. 115.
575 Outline of a New System of Logic, 1827, p. 117.
576 Porphyrii Isagoge, Caput ii. 24.
577 Jevons, Elementary Lessons in Logic, p. 104.
578 Chrestomathia; being a Collection of Papers, &c. London, 1816, Appendix V.
579 The Classification of the Sciences, &c., 3rd edit. p. 7. Essays: Scientific, Political, and Speculative, vol. iii. p. 13.
580 Owen, Essay on the Classification and Geographical Distribution of the Mammalia, p. 20.
581 Dana’s Mineralogy, vol. i. p. 123; quoted in Watts’ Dictionary of Chemistry, vol. ii. p. 166.
582 Instructions for the Discrimination of Minerals by Simple Chemical Experiments, by Franz von Kobell, translated from the German by R. C. Campbell. Glasgow, 1841.
583 Edition of 1866, p. lxiii.
584 Philosophia Botanica (1770), § 154, p. 98.
585 Philosophical Magazine, 3rd Series (1845), vol. xxvi. p. 522. See also De Morgan’s evidence before the Royal Commission on the British Museum in 1849, Report (1850), Questions, 5704*-5815*, 6481–6513. This evidence should be studied by every person who wishes to understand the elements of Bibliography.
586 English Cyclopædia, Arts and Sciences, vol. v. p. 233.
587 Swainson, “Treatise on the Geography and Classification of Animals,” Cabinet Cyclopædia, p. 201.
588 Darwin, Fertilisation of Orchids, p. 159.
589 Descent of Man, vol. i. p. 214.
590 Laws of Botanical Nomenclature, p. 16.
591 Waterhouse, quoted by Woodward in his Rudimentary Treatise of Recent and Fossil Shells, p. 61.
592 Bentham’s Handbook of the British Flora (1866), p. 25.
593 Philosophia Botanica (1770), § 157, p. 99.
594 Ibid. § 159, p. 100.
595 Amœnitates Academicæ (1744), vol. i. p. 70. Quoted in Edinburgh Review, October 1868, vol. cxxviii. pp. 416, 417.
596 Descent of Man, vol. i. p. 228.
597 Agassiz, Essay on Classification, p. 219.
598 Ibid. p. 249.
599 Philosophia Botanica, § 155, p. 98.
600 Laws of Botanical Nomenclature, by Alphonse Decandolle, translated from the French, 1868, p. 19.
601 Darwin, The Variation of Animals and Plants, vol. ii. pp. 293, 359, &c.; quoting Paget, Lectures on Pathology, 1853, pp. 152, 164.
602 Ibid. vol. ii. p. 372.
603 Théorie Analytique des Probabilités, quoted by Babbage, Ninth Bridgewater Treatise, p. 173.
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.
607 Ninth Bridgewater Treatise, p. 140.
608 Ibid. pp. 34–43.
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.
613 Report of the British Association (1852), Report of Sections, p. 12.
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.
616 Familiar Lectures on Scientific Subjects, p. 458.
617 Philosophical Magazine, 3rd Series, vol. xxvi. p. 406.
618 History of the Theory of Probability, p. 398.
619 Trigonometry and Double Algebra, chap. ix.
620 Agassiz, Essay on Classification, p. 75.
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
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