Charles Babbage

(From The Victorian Web).

The volume here presented to the public does not form a part of that series of works composed at the desire of the trustees who directed the application of the bequest of £8000, by the late Earl of Bridgewater, for the purpose of advancing arguments in favor of Natural Religion. I have, however, thought, that in furthering the intentions of the testator, by publishing some reflections on that subject, I might be permitted to connect with them a title which has now become familiarly associated, in the public mind, with the evidences in favor of Natural Religion.

The Bridgewater Treatises were restricted by the founder to the subject of Natural Religion; and I had intended not to have deviated from their example. In the single instance in which the question of miracles has been discussed, I was led so irresistibly, by the very nature of the illustrations employed in the former argument, to the view there proposed, that I trust to being excused for having ventured one step beyond the strict limits of that argument, by entering on the first connecting link between natural religion and revelation.

The same argument will produce very various degrees of conviction on different minds; and much of this difference will depend on the extent of previous information, and on the strength of the reasoning faculty in those to whom the argument is addressed. To the great variety, therefore, of the illustrations which have been adduced in proof of design and of benevolence in the works of the Creator, there can be no objection. In truth, to the cultivated eye of science, the origin and consequences of the mightiest hurricane, as well as those of the smallest leaf it scatters in its course, equally lead to the inference of a designing power, the more irresistibly the more extensive the knowledge which is brought to bear on those phenomena.

One of the chief defects of the Treatises above referred to appears to me to arise from their not pursuing the argument to a sufficient extent. When a multitude of apparently unconnected facts is traced up to some common principle, we feel spontaneously an admiration for him who has explained to us the connection; and if, advancing another stage in the investigation, he prove that other facts, apparently at variance with that principle, are not merely no exceptions, but are themselves inevitable consequences of its application, our admiration of the principle, and our respect for its discoverer, are still further enhanced.

But if this respect and admiration are yielded to the mere interpreter of Nature's laws, how much more exalted must those sentiments become when applied to the Being who called such principles into living existence by creating matter subservient to their dominion—whose mind, intimately cognizant of the remotest consequences of the present as well as of all other laws, decreed existence to that one alone, which should comprehend within its grasp the completion of its destiny— which should require no future intervention to meet events unanticipated by its author, in whose omniscient mind we can conceive no infirmity of purpose—no change of intention!

The object of these pages, as of the Bridgewater Treatises, is to show that the power and knowledge of the great Creator of matter and of mind are unlimited. Deeply engaged in those other pursuits from which my chief arguments are drawn, I regret the impossibility of bestowing on their full development that time and attention which the difficulty and importance of the subject equally deserve; and in committing these fragments to the press, perhaps in too condensed a form, I wish them to be considered merely as suggestions intended to direct the reader's attention to lines of argument which appear to me new, and to views of nature which appear more magnificent, than those with which I was previously acquainted. Probably I should not have been induced to place my reflections on the subject before the public, had I not, in common with other cultivators of the more abstract branches of mathematical science, felt that a prejudice, which I had believed to have been long eradicated from every cultivated mind, had lately received support, at least to a certain extent, from a chapter in the first of the Bridgewater Treatises; and in a still greater degree, from a work of a far different order—one, however, which derived its only claim to notice from the circumstance of its appearing under the sanction of the University of Oxford.

The prejudice to which I allude is, that the pursuits of science are unfavorable to religion.

There are two classes of men most deeply impressed with the conviction of the very limited extent of human knowledge—those whose contracted information renders them eminent examples of the fact, and those whose wide grasp of many of its profoundest branches has taught them, by lengthened experience, that each accession to their stock but enables them to view a larger portion of its illimitable field. Those who belong to the first of these classes must acquire the alphabet of science, in order to understand knowledge, and the elements of modesty, to use it with dignity. When they have thus graduated in the "infant school" of philosophy, they may perhaps understand the argument, and perchance be worthy of a reply,—but not till then.

In that chapter of the first Bridgewater Treatise to which I have referred, the charge seems not even to be limited to those who pursue that branch of science which is conversant with the properties of pure number, and with abstractions of a like nature, but applies to all who cultivate deductive processes of reasoning.

It is maintained by the author, that long application to such inquiries disqualifies the mind from duly appreciating the force of that kind of evidence which alone can be adduced in favor of Natural Theology.

"We may thus, with the greatest propriety, deny to the mechanical philosophers and mathematicians of recent times any authority with regard to their views of the administration of the universe; we have no reason whatever to expect from their speculations any help, when we ascend to the first cause and supreme ruler of the universe. But we might perhaps go farther, and assert that they are in some respects less likely than men employed in other pursuits, to make any clear advance towards such a subject of speculation."—Bridgewater Treatise) by the Rev. WM. WHEWELL, p. 334.

Admitting, for the sake of argument, that there have been individuals, possessed of high intellectual powers, successfully devoted to those subjects, who have arrived by reasoning at conclusions respecting the First Cause, totally opposite to those entertained by Mr. Whewell and myself, I should still be very reluctant to endeavor to invalidate the influence of their conclusions, by any inquiry either into their intellectual or their moral character. Reasoning is to be combated and refuted by reasoning alone. Any endeavor to raise a prejudice, or throw the shadow of an imputation, either implies the existence of some latent misgiving in the minds of those who employ such weapons, or is a tacit admission that the question is beyond the grasp of one at least of the debaters.

Who that has studied their works ever dreamed of inquiring into the moral or intellectual character of Euclid or Archimedes, for the purpose of confirming or invalidating his belief in their conclusions? Who that possesses confidence in his own reason, justified by a laborious cultivation and successful exercise of that faculty, fails to anatomize and refute the arguments, rather than analyze the mental or moral habits of those from whom he differs?

The only case in which such extraneous matters can be fairly called in, is when facts are stated resting on testimony. Then it is not only just, but it is necessary for the sake of truth, to inquire into the habits of mind of him by whom they are adduced;— whether he possesses sufficient talent and precision to enable him to state precisely what his senses convey to him, and nothing more; or, if he receive information from others, whether he is credulous or cautious. In both cases, it is necessary to inquire into moral feelings, in order to be assured that there is no willful misstatement in the groundwork of his reasoning. And even when this is well established, it is still necessary to inquire whether he had any personal, professional, or pecuniary interest which may insensibly have influenced his mind in one direction.

Such I conceive to be the sound distinction between those branches of knowledge resting on facts open to the observation of all, supported by reasoning addressed to the understandings of all,— and those other branches in which reasoning is mixed up with testimony. In the former, the argument is every thing—the character nothing: in the latter, the character must be sifted as well as the arguments.

Feeling convinced that the truths of Natural Religion rest on foundations far stronger than those of any human testimony; that they are impressed in indelible characters, by almighty power, on every fragment of the material world, I cannot but regret that reflections should have been made, in connection with this subject, calculated to throw the least shadow of doubt on evidence otherwise irresistible.

As, however, these views of the nature of the question may not bring that conviction to other minds, which they do to my own, and as one of the disturbing forces which act on our minds has been strongly put forward, it is but justice to state the whole of them. It requires but little insight into man's heart to perceive that profession and professional advancement — that power and wealth — have a far more frequent and more effective influence on his judgment than any mental habits he may be supposed to have cultivated.

It may be right then to state, that the author of these pages has always been an ardent but not an exclusive cultivator of some of the more abstract branches of mathematical science. In pursuing one of those inquiries, amongst the most recondite and apparently the most removed from any practical application, he was struck with the bearing of some of the resets which presented themselves, on the question of Natural Religion; and these he has endeavored to place before the reader, in the following pages.

The author belongs to no profession in which he can hope for advancement, if he successfully advocate one side of the question, or in which his prospects can be injured by candidly stating any arguments on the other. He has not been invited by men high in the State, and deservedly respected, to strengthen that great basis which precedes all revelation, and on which it must all rest. Nor has any sum of money been assigned to him, that, whatever the mercantile success or failure of the present volume may be, he shall, on its publication, reap a large pecuniary reward. Having chosen a career to which the institutions of the country hold out none of those great prizes that stimulate professional exertions, and which constrain men to yield a certain degree of deference to the opinions, sound or unsound, of their countrymen, he has, on the one hand, nothing to hope from their approbation, and, on the other, is equally exempt from any dread of their censure; and, had his conviction been as strongly opposed to the doctrines this Fragment advocates, as it is in their favor, he would, had a fit occasion presented itself, fearlessly have laid before the world the arguments which had forced his mind to that conviction.

In conclusion, I have to express to my fellow-laborers in the cause, my hope that they will put no unkind interpretation on these remarks, which, founded on principles of human nature, are necessarily of general application; that they will see that motives alien, in my own opinion, to the subject, having been once introduced, candor to those who differ from us, as well as a deference to truth itself, compelled me to state them fully.

CHARLES BABBAGE.
DORSET-STREET,
MANCHESTER-SQUARE, April 1837.

I have again read, with much attention, the chapters in Mr. Whewell's Bridgewater Treatise, which bear upon the question of the effect of the pursuits of science on our belief in natural religion, and I confess that I am unable to alter the opinion I have already expressed upon that subject,—that they give support to those who maintain that the pursuits of science are in general unfavorable to religion. Of the injustice of that opinion and of the individual injury which it occasionally inflicts, additional evidence has been presented to me, since the publication of the First Edition.

In endeavoring to understand the meaning and spirit of the author, I find it difficult to interpret such passages as the following, which is stated to embody the substance of his opinions:—

"If the mathematician set out on religious reasonings, thinking that his mathematical knowledge alone must bring him into a nearer proximity to his Maker and Master, he will, I fear, find that the road is interrupted by a wide chasm, and he may, perhaps, turn back frustrated and hopeless. It is only by rising above his mathematics and his physics;—by recognizing the utter dissimilarity of moral and religious grounds of belief, from mathematical and physical reasonings upon established laws of nature;—that he can make his way to the conviction of a moral constitution and providential government of the world; and if the mathematical or physical philosopher so habituate his mind, that it is difficult for him thus to elevate himself into a higher region than that of mathematical proof and physical consequence, I cannot but think he does damage to his power of judging on those other subjects."

This passage is one of those which is likely to be misunderstood, and which may be adduced by others in the support of views which it is impossible to suppose Mr. Whewell to entertain. If, by "rising above his mathematics and physics," it is meant, that inquiry into the relation of man to his Maker, is of more importance to his welfare than those other subjects, then it is a proposition which scarcely requires to be asserted, because it has never been denied. Even the atheist, who has arrived, by reasoning, at his desolate conclusion, would not fail to admit its truth, by attending to any new argument which might be proposed against his creed. But if it is meant, that there is a "higher region" of evidence than that of "mathematical proof and physical consequence," then it is in my opinion utterly and completely erroneous; and as I am confident this erroneous light will be that in which the statement will be understood by many, I think it necessary to state distinctly what appears to me the relative position of the subjects in discussion.

First, The truths of pure mathematics are necessary truths; they are of such a nature, that to suppose the reverse, involves a contradiction.

Secondly, The laws of nature, on which physical reasonings are founded, although some of them are considered as necessary truths, depend, in many instances, on the testimony of our senses. These derive their highest confirmation from the aid of pure mathematics, by which innumerable consequences, previously unobserved, are proved to result from them.

Thirdly, The truths of natural religion rest also on the testimony of our external senses, but united with that internal consciousness of intention or design which we experience in our own breast, and from which we infer similar powers in other beings. Many of the facts on which the conclusions of natural religion are founded, derive their chief importance from the aid supplied by the united power of the two former classes, and the amount and value of this support will be enlarged with the advance of those sciences. Fourthly, Revealed religion rests on human testimony; and on that alone. Its first and greatest support arises from natural religion. I have endeavored in one chapter of the present volume to show, that, notwithstanding the weakening effect of transmission upon testimony, a time may arrive when, by the progress of knowledge, internal evidence of the truth of revelation may start into existence with all the force that can be derived from the testimony of the senses.

The first class of truths then (those of Pure Mathematics) appears to rest on necessity. The second, (the Laws of Nature,) on necessity and our external senses. The third, (those of Natural Religion,) on our external senses and internal consciousness. The last, (those of Revelation,) on human testimony. If they admit of any classification, as subjects having a common resemblance, or as possessing different degrees of evidence, I have placed them in the only order which, in my opinion, is consistent with truth; convinced that it is more injurious to religion to overrate, than to undervalue the cogency of the evidence on which it rests.

CHARLES BABBAGE.
DORSET STREET,
MANCHESTER SQUARE, Dec. 26, 1837.

Chapter I. Nature of the Argument

The notions we acquire of contrivance and design arise from comparing our observations on the works of other beings with the intentions of which we are conscious in our own undertakings. We take the highest and best of human faculties, and, exalting them in our imagination to an unlimited extent, endeavor to attain an imperfect conception of that Infinite Power which created every thing around us. In pursuing this course, it is evident that we are liable to impress upon the notion of Deity thus shadowed out, many traces of those imperfections in our own limited faculties which are best known to those who have most deeply cultivated them. It is also evident that all those discoveries which arm human reason with new power, and all additions to our acquaintance with the material world, must from time to time render a revision of that notion necessary. The present seems to be a fit occasion for such a revision.

Many excellent and religious persons not deeply versed in what they mistakenly call "human knowledge" but which is in truth the interpretation of those laws that God himself has impressed on his creation, have endeavored to discover proofs of design in a multitude of apparent adaptations of means to ends, and have represented the Deity as perpetually interfering, to alter for a time the laws he had previously ordained; thus by implication denying to him the possession of that foresight which is the highest attribute of omnipotence. Minds of this order, insensible of the existence of that combining and generalizing faculty which gives to human intellect its greatest development, and tied down by the trammels of their peculiar pursuits, have in their mistaken zeal not perceived their own unfitness for the mighty task, and have ventured to represent the Creator of the universe as fettered by the same infirmities as those by which their own limited faculties are subjugated. To causes of this kind must in some measure be attributed an opinion which has been industriously spread, that minds highly imbued with mathematical knowledge are disqualified, by the possession of that knowledge, and by the habits of mind produced during its acquisition, from rightly appreciating the works of the Creator.

At periods and in countries in which the knowledge of the priests exceeded that of the people, science has always been held up by the former class as an object of regard, and its crafty possessors have too frequently defiled its purity by employing their knowledge for the delusion of the people. On the other hand, at times and in countries in which the knowledge of the people has advanced beyond that of the priesthood, the ministers of the temple have too often been afraid of the advance of knowledge, and have threatened with the displeasure of the Almighty those engaged in employing the faculties he has bestowed on the study of the works he has created. At the present period, when knowledge is so universally spread that neither class is far in advance of the other,—when every subject is submitted to unbounded discussion,—when it is at length fully acknowledged that truth alone can stand unshaken by perennial attacks, and that error, though for centuries triumphant, must fall at last, and leave behind no ashes from which it may revive, the authority of names has but little weight: facts and arguments are the basis of creeds, and convictions so arrived at are the more deeply seated, and the more enduring, because they are not the wild fancies of passion or of impulse, but the deliberate results of reason and reflection.

It is a condition of our race that we must ever wade through error in our advance towards truth; and it may even be said that in many cases we exhaust almost every variety of error before we attain the desired goal. But those truths, once reached by such a course, are always most highly valued; and when, in addition to this, they have been exposed to every variety of attack which splendid talents quickened into energy by the keen perception of personal interests can suggest,—when they have revived undying from unmerited neglect; when the anathema of spiritual, and the arm of secular power have been found as impotent in suppressing, as arguments were in refuting them, then they are indeed irresistible. Thus tried and thus triumphant in the fiercest warfare of intellectual strife, even the temporary interests and furious passions which urge on the contest, contribute in no small measure to establish their value, and thus to render these truths the permanent heritage of our race.

Viewed in this light, the propagation of an error, although it may be unfavorable or fatal to the temporary interest of an individual, can never be long injurious to the cause of truth. It may, at a particular period, retard its progress for a while, but it repays the transitory injury by a benefit as permanent as the duration of the truth to which it was opposed. These reflections are offered for the purpose of proving that the toleration of the fullest discussion is most advantageous to truth. They are not offered as the apology for error; and whilst it is admitted that every person who willfully puts forward arguments the soundness of which he doubts, incurs a deep responsibility, it is some satisfaction to reflect that the delay likely to be thus occasioned to the great cause can be but small; and that those who in sincerity of heart maintain arguments which a more advanced state of knowledge shall prove to be erroneous, may yet ultimately contribute, by their very publication, to the speedier establishment of truth.

Chapter II. Argument in Favor of Design from the Changing of Laws in Natural Events

The estimate we form of the intellectual capacity of our race, is founded on an examination of those productions which have resulted from the loftiest flights of individual genius, or from the accumulated labors of generations of men, by whose long-continued exertions a body of science has been raised up, surpassing in its extent the creative powers of any individual, and demanding for its development a length of time, to which no single life extends.

The estimate we form of the Creator of the visible world rests ultimately on the same foundation. Conscious that we each of us employ, in our own productions, means intended to accomplish the objects at which we aim, and tracing throughout the actions and inventions of our fellow-creatures the same intention,— judging also of their capacity by the fit selection they make of the means by which they work, we are irresistibly led, when we contemplate the natural world, to attempt to trace each existing fact presented to our senses to some precontrived arrangement, itself perhaps the consequence of a yet more general law; and where the most powerful aids by which we can assist our limited faculties fail in enabling us to detect such connections, we still, and not the less, believe that a more extended inquiry, or higher powers, would enable us to discover them.

The greater the number of consequences resulting from any law, and the more they are foreseen, the greater the knowledge and intelligence we ascribe to the being by which it was ordained. In the earlier stages of our knowledge, we behold a multitude of distinct laws, all harmonizing to produce results which we deem beneficial to our own species: as science advances, many of these minor laws are found to merge into some more general principles; and with its higher progress these secondary principles appear, in their turn, the mere consequences of some still more general law. Such has been the case in two of the most curious and most elaborately cultivated branches of human knowledge, the sciences of astronomy and optics. All analogy leads us to infer, and new discoveries continually direct our expectation to the idea, that the most extensive laws to which we have hitherto attained, converge to some few simple and general principles, by which the whole of the material universe is sustained, and from which its infinitely varied phenomena emerge as the necessary consequences. [See Note A in the Appendix.]

To illustrate the distinction between a system to which the restoring hand of its contriver is applied, either frequently or at distant intervals, and one which had received at its first formation the impress of the will of its author, foreseeing the varied but yet necessary laws of its action throughout the whole of its existence, we must have recourse to some machine, the produce of human skill. But far as all such engines must ever be placed at an immeasurable interval below the simplest of Nature's works, yet, from the vastness of those cycles which even human contrivance in some cases unfolds to our view, we may perhaps be enabled to form a faint estimate of the magnitude of that lowest step in the chain of reasoning, which leads us up to Nature's God.

The illustration which I shall here employ will be derived from the results afforded by the Calculating Engine [the reader will find a short account of this engine in the Appendix, Note B]; and this I am the more disposed to use, because my own views respecting the extent of the laws of Nature were greatly enlarged by considering it, and also because it incidentally presents matter for reflection on the subject of inductive reasoning. Nor will any difficulty arise from the complexity of that engine; no knowledge of its mechanism, nor any acquaintance with mathematical science, are necessary for comprehending the illustration; it being sufficient merely to conceive that computations of great complexity can be effected by mechanical means.

Let the reader imagine that such an engine has been adjusted; that it is moved by a weight; and that he sits down before it, and observes a wheel, which moves through a small angle round its axis, at short intervals, presenting to his eye, successively, a series of numbers engraved on its divided circumference. Let the figures thus seen be the series of natural numbers, 1, 2, 3, 4, 5, &c., each of which exceeds its immediate antecedent by unity.

Now, reader, let me ask how long you will have counted before you are firmly convinced that the engine, supposing its adjustments to remain unaltered, will continue whilst its motion is maintained, to produce the same series of natural numbers? Some minds perhaps are so constituted, that after passing the first hundred terms they will be satisfied that they are acquainted with the law. After seeing five hundred terms, few will doubt; and after the fifty-thousandth term the propensity to believe that the succeeding term will be fifty thousand and one, will be almost irresistible. That term will be fifty thousand and one: the same regular succession will continue; the five-millionth and the fifty-millionth term will still appear in their expected order; and one unbroken chain of natural numbers will pass before your eyes, from one up to one hundred million. True to the vast induction which has thus been made, the next succeeding term will be one hundred million and one; but after that the next number presented by the rim of the wheel, instead of being one hundred million and two, is one hundred million ten thousand and two. The whole series from the commencement being thus:—

The law which seemed at first to govern this series fails at the hundred million and second term. That term is larger than we expected, by 10,000. The next term is larger than was anticipated, by 30,000, and the excess of each term above what we had expected forms the following table:—

 10,000
 30,000
 60,000
100,000

being, in fact, the series of triangular numbers, each multiplied by 10,000.

The numbers 1, 3, 6, 10, 15, 21, 28, &c. are formed by adding the successive terms of the series of natural numbers thus;

1 = 1

1+2 = 3. 1+2 + 3 = 6.

1 + 2 + 3 + 4 = 10, &c.

They are called triangular numbers, because a number of points corresponding to any term can always be placed in the form of a triangle, for instance:—

. . . 3

. . . . . . 6

. . . . . . . . . . 10

If we still continue to observe the numbers presented by the wheel, we shall find, that for a hundred, or even for a thousand terms, they continue to follow the new law relating to the triangular numbers; but after watching them for 2761 terms, we find that this law fails in the case of the 2762d term.

If we continue to observe, we shall discover another law then coming into action, which also is dependent, but in a different manner, on triangular numbers. This will continue through about 1430 terms, when a new law is again introduced, which extends over about 950 terms; and this too, like all its predecessors, fails, and gives place to other laws, which appear at different intervals.

Now it must be remarked, that the law that each number presented by the Engine is greater by unity than the preceding number, which law the observer had deduced from an induction of a hundred million instances, was not the true law that regulated its action; and that the occurrence of the number 100,010,002 at the 100,000,002d term, was as necessary a consequence of the original adjustment, and might have been as fully foreknown at the commencement, as was the regular succession of any one of the intermediate numbers to its immediate antecedent. The same remark applies to the next apparent deviation from the new law, which was founded on an induction of 2761 terms, and to all the succeeding laws; with this limitation only—that whilst their consecutive introduction at various definite intervals is a necessary consequence of the mechanical structure of the engine, our knowledge of analysis does not yet enable us to predict the periods at which the more distant laws will be introduced.

Such are some of the facts which, by a certain adjustment of the Calculating Engine, would be presented to the observer. Now, let him imagine another engine, offering to the eye precisely the same figures in the same order of succession; but let it be necessary for the maker of that other engine, previously to each apparent change in the law, to make some new adjustment in the structure of the engine itself, in order to accomplish the ends proposed. The first engine must be susceptible of having embodied in its mechanical structure, that more general law of which all the observed laws were but isolated portions,—a law so complicated, that analysis itself, in its present state, can scarcely grasp the whole question. The second engine might be of far simpler contrivance; it must be capable of receiving the laws impressed upon it from without, but is incapable, by its own intrinsic structure, of changing, at definite periods, and in unlimited succession, those laws by which it acts. Which of these two engines would, in the reader's opinion, give the higher proof of skill in the contriver? He cannot for a moment hesitate in pronouncing that that for which, after its original adjustment, no superintendance is required, displays far greater ingenuity than that which demands, at every change in its law, the direct intervention of its contriver.

The engine we have been considering is but a very small portion (about fifteen figures) of a much larger one, which was preparing, and is partly executed; it was intended, when completed, that it should have presented at once to the eye about one hundred and thirty figures. In that more extended form which recent simplifications have enabled me to give to machinery constructed for the purpose of making calculations, it will be possible, by certain adjustments, to set the engine so that it shall produce the series of natural numbers in regular order, from unity up to a number expressed by more than a thousand places of figures. At the end of that term, another and a different law shall regulate the succeeding terms; this law shall continue in operation perhaps for a number of terms, expressed perhaps by unity, followed by a thousand zeros, or 10¹⁰⁰⁰; at which period a third law shall be introduced, and, like its predecessors, govern the figures produced by the engine during a third of those enormous periods. This change of laws might continue without limit; each individual law being destined to govern for millions of ages the calculations of the engine, and then give way to its successor to pursue a like career. [It has been supposed that ten turns of the handle of the calculating engine might be made in a minute, or about five hundred and twenty-six millions in a century. As in this case, each turn would make a calculation, after the lapse of a million of centuries, only the fifteenth place of figures would have been reached.]

Thus a series of laws, each simple in itself, successively spring into existence, at distances almost too great for human conception. The full expression of that wider law, which comprehends within it this unlimited sequence of minor consequences, may indeed be beyond the utmost reach of mathematical analysis: but of one remarkable fact, however, we are certain—that the mechanism brought into action for the purpose of changing the nature of the calculation from the production of the merest elementary operations into those highly complicated ones of which we speak, is itself of the simplest kind.

In contemplating the operations of laws so uniform during such immense periods, and then changing so completely their apparent nature, whilst the alterations are in fact only the necessary consequences of some far higher law, we can scarcely avoid remarking the analogy which they bear to several of the phenomena of nature.

The laws of animal life which regulate the caterpillar, seem totally distinct from those which, in the subsequent stage of its existence, govern the butterfly. The difference is still more remarkable in the transformations undergone by that class of animals which spend the first portion of their life beneath the surface of the waters, and the latter part as inhabitants of air. It is true that the periods during which these laws continue to act are not, to our senses, enormous, like the mechanical ones above mentioned; but it cannot be doubted that, immeasurably more complex as they are, they were equally foreknown by their Author: and that the first creation of the egg of the moth, or the libellula, involved within its contrivance, as a necessary consequence, the whole of the subsequent transformations of every individual of their respective races.

In turning our views from these simple results of the juxtaposition of a few wheels, it is impossible not to perceive the parallel reasoning, which may be applied to the mighty and far more complex phenomena of nature. To call into existence all the variety of vegetable forms, as they become fitted to exist, by the successive adaptations of their parent earth, is undoubtedly a high exertion of creative power. When a rich vegetation has covered the globe, to create animals adapted to that clothing, which, deriving nourishment from its luxuriance, shall gladden the face of nature, is not only a high but a benevolent exertion of creative power. To change, from time to time, after lengthened periods, the races which exist, as altered physical circumstances may render their abode more or less congenial to their habits, by allowing the natural extinction of some races, and supplying by a new creation others more fitted to occupy the place previously abandoned, is still but the exercise of the same benevolent power. To cause an alteration in those physical circumstances,—to add to the comforts of the newly-created animals,—all these acts imply power of the same order, a perpetual and benevolent superintendence, to take advantage of altered circumstances, for the purpose of producing additional happiness.

But, to have foreseen, at the creation of matter and of mind, that a period would arrive when matter, assuming its prearranged combinations, would become susceptible of the support of vegetable forms; that these should in due time themselves supply the pabulum of animal existence; that successive races of giant forms or of microscopic beings should at appointed periods necessarily rise into existence, and as inevitably yield to decay; and that decay and death—the lot of each individual existence—should also act with equal power on the races which they constitute; that the extinction of every race should be as certain as the death of each individual, and the advent of new genera be as inevitable as the destruction of their predecessors;—to have foreseen all these changes, and to have provided, by one comprehensive law, for all that should ever occur, either to the races themselves, to the individuals of which they are composed, or to the globe which they inhabit, manifests a degree of power and of knowledge of a far higher order.

The vast cycles in the geological changes that have taken place in the earth's surface, of which we have ample evidence, offer another analogy in nature to those mechanical changes of law from which we have endeavored to extract a unit sufficiently large to serve as an imperfect measure for some of the simplest works of the Creator.

The gradual advance of Geology, during the last twenty years, to the dignity of a science, has arisen from the laborious and extensive collection of facts, and from the enlightened spirit in which the inductions founded on those facts have been deduced and discussed. To those who are unacquainted with this science, or indeed to any person not deeply versed in the history of this and kindred subjects, it is impossible to convey a just impression of the nature of that evidence by which a multitude of its conclusions are supported:—evidence in many cases so irresistible, that the records of the past ages, to which it refers, are traced in language more imperishable than that of the historian of any human transactions; the relics of those beings, entombed in the strata which myriads of centuries have heaped upon their graves, giving a present evidence of their past existence, with which no human testimony can compete.

It is found that each additional step, in the grouping together of the facts of geology, confirms the view that the changes of our planet, since it has been the abode of man, is but as a page in the massive volumes of its history, every leaf of which, written in the same character, conveys to the decipherer the idea of a succession of the same causes acting with varying intensity, through unequal but enormous periods, each period apparently distinguished by the coming in or going out of new subsidiary laws, yet all submitted to some still higher condition, which has stamped the mark of unity on the series, and points to the conclusion that the minutest changes, as well as those transitions apparently the most abrupt, have been throughout all time the necessary, the inevitable consequences of some more comprehensive law impressed on matter at the dawn of its existence.

Chapter III. Argument to show that the Doctrines in the preceding Chapter do not lead to Fatalism

If all the combinations and modifications of matter can be supposed to be traced up to one general and comprehensive law, from which every visible form, both in the organic and inorganic world flows, as the necessary consequence of the first impression of that law upon matter, it might seem to follow that Fate or Necessity governs all things, and that the world around us may not be the result of a contriving mind working for a benevolent purpose.

Such, possibly, may be the first impression of this view of the subject; but it is an erroneous view,—one of those, perhaps, through which it is necessary to pass, in order to arrive at truth. Let us, in order to obtain more correct views upon this point, briefly review the labor which the human race has expended, in attaining the limited knowledge we possess. For about six thousand years man has claimed the earth as his heritage, and asserted his dominion over all other beings endued with life; yet, during a large portion of that period, how small comparatively has been his mental improvement! Until the invention of printing, the mass of mankind were in many respects almost the creatures of instinct. It is true, the knowledge possessed by each generation, instead of being the gift of Nature, was derived from the instruction of their predecessors; but, how little were those lessons improved by repeated communication! Transmitted most frequently by unenlightened instructors, they might lose, but could rarely gain in value.

Before the invention of printing, accidental position determined the opinions and the knowledge of the great mass of mankind. Oral information being almost the only kind accessible, each man shared the opinions of his kindred and neighbors; and truth, which is ever most quickly and most surely elicited by discussion, lost all those advantages which diversity of opinion always produces for it. The minds of individual men, however powerful, could address themselves only to a very small portion of their fellow men; their influence was limited by space and restricted by time; their highest powers were not stimulated into action by the knowledge that their reasonings could have effect where their voices were unheard, by the conviction that the truths which they arrived at, and the discoveries they made, would extend beyond their country and survive their age.

But, since the invention of printing, how different has been the position of mankind! The nature of the instruction no longer depends entirely on the knowledge of the personal instructor. The village schoolmaster communicates to his pupils the power of using an instrument by which not merely the best of their living countrymen, but the greatest and wisest men of all countries and all times, may become their instructors. Even the elementary writings through which this art is taught, give to the pupil, not the sentiments of the teacher, but those which the public opinion of his countrymen esteems most fit for the beginner in knowledge. Thus the united opinions of multitudes of human minds are brought to bear even upon seemingly unimportant points. If such is the effect of the invention of printing upon ordinary minds, its influence over those more highly endowed is far greater. To them the discussion of the conflicting opinions of different countries and distant ages, and the establishment of new truths, present a field of boundless and exalted ambition. Advancing beyond the knowledge of their neighbors and countrymen, they may be exposed to those prejudices which result from opinions long stationary; but encouraged by the approbation of the greatest of other nations, and the more enlightened of their own,— knowing that time alone is wanting to complete the triumph of truth, they may accelerate the approaching dawn of that day which shall pour a flood of light over the darkened intellects of their thankless countrymen—content themselves to exchange the hatred they experience from the honest and the dishonest intolerance of their contemporaries, for that higher homage, alike independent of space and of time, which their memory will for ever receive from the good and the gifted of all countries and all after ages.

Until printing was very generally spread, civilization scarcely advanced by slow and languid steps; since this art has become cheap, its advances have been unparalleled, and its rate of progress vastly accelerated.

It has been stated that the civilization of the Western World has resulted from its being the seat of the Christian religion: but however much the mild tenor of its doctrines is calculated to assist in producing such an effect, that religion cannot but be injured by an unfounded statement. It is to the easy and cheap methods of communicating thought from man to man, which enable a country to sift, as it were, its whole people, and to produce, in its science, its literature, and its arts, not the brightest efforts of a limited class, but the highest exertions of the most powerful minds among a whole community;—it is this which has given birth to the wide-spreading civilization of the present day, and which promises a futurity yet more prolific. Whoever is acquainted with the present state of science and the mechanical arts, and looks back over the inventions and civilization which the fourteen centuries subsequent to the introduction of Christianity have produced, and compares them with the advances made during the succeeding four centuries following the invention of printing, will have no doubt as to the effective cause.

It is during these last three or four centuries, that man, considered as a species, has commenced the development of his intellectual faculties—that he has emerged from a position in which he was almost the creature of instinct, to a state in which every step in advance facilitates the progress of his successors. During the first period, arts were discovered by individuals, and lost to the race; in the latter, the diffusion of thought has enabled the reasoning of one class to unite with the observations of another, and the most advanced point of one generation to become the starting post of the next. It is during this portion of our history that man has become acquainted with his real position in the universe—that he has measured the distance from that which is to us the great fountain of light and heat—that he has traced the orbits of earth's sister spheres, and calculated the paths of all their dependent worlds—that he has arrived at the knowledge of a law which appears to govern all matter, and whose remotest consequences, if first traced by his telescope, are found to have been written in his theory; or, if first predicted by his theory, are verified by his observations.

Simple as the law of gravity now appears, and beautifully in accordance with all the observations of past and of present times, consider what it has cost of intellectual study. Copernicus, Galileo, Kepler, Euler, Lagrange, Laplace, all the great names which have exalted the character of man, by carrying out trains of reasoning unparalleled in every other science; these, and a host of others, each of whom might have been the Newton of another field, have all labored to work out, the consequences which resulted from that single law which he discovered. All that the human mind has produced—the brightest in genius, the most persevering in application, has been lavished on the details of the law of gravity.

Had that law been other than it is—had it been, for example, the inverse cube of the distance, it would still have required an equal expense of genius and of labor to have worked out its details. But, between the laws represented by the inverse square, and the inverse cube of the distance, there are interposed an infinite number of other laws, each of which might have been the basis of a system requiring the most extensive knowledge to trace out its consequences. Again, between every law which can be expressed by whole numbers, whether it be direct or inverse, an infinity of others can still be interposed. All these might be combined by two, by three, or in any other groups, and new systems might be imagined [even beyond this, every law so imagined might be interrupted by any discontinuous function; and thus be made to agree, for any period, with laws of simpler form, and yet deviate, in one single, or in a certain limited number of cases, and then agree with it for ever], submitted to such combinations. Thus, another infinity of laws, of a far higher order—in fact, of an infinitely higher order — might again be added to the list. And this might still be increased by all the other combinations, of which such laws admit, besides that by addition, to which we have already alluded, thus forming an infinity itself of so high an order, that it is difficult to conceive. Man has, as yet, no proof of the impossibility of the existence of any of these laws. Each might, for any reason we can assign, be the basis of a creation different from our own.

It is at this point that skill and knowledge re-enter the argument, and banish for ever the dominion of chance. The Being who called into existence this creation, of which we are parts, must have chosen the present form, the present laws, in preference to the infinitely infinite variety which he might have willed into existence. He must have known and foreseen all, even the remotest consequences of every one of those laws, to have penetrated but a little way into one of which has exhausted the intellect of our whole species.

If such is the view we must take of the knowledge of the Creator, when contemplating the laws of inanimate matter—laws into whose consequences it has cost us such accumulated labor to penetrate—what language can we speak, when we consider that the laws which connect matter with animal life may be as infinitely varied as those which regulate material existence? The little we know, might, perhaps, lead us to infer a far more unlimited field of choice. The chemist has reduced all the materials of the earth with which we are acquainted, to about fifty simple bodies; but the zoologist can make no such reductions in his science. He claims for one scarcely noticed class — that of intestinal parasites — about thirty thousand species; and, not to mention the larger classes of animals, who shall number the species of infusoria in living waters, still less those which are extinct, and whose scarcely visible relics are contained within the earth, in almost mountain masses. [Professor Ehrenberg, of Berlin, has discovered that the tripoli employed in that city for polishing metals, which is dug up at Bilin, in Bohemia, consists almost entirely of the siliceous remains of infusoria, of a species so minute, that about 41,000 millions of them weigh 220 grains, and occupy the space of a cubic inch. The reader will find a translation of the highly interesting papers of Professor Ehrenberg, in the third number of the " Scientific Memoirs," published by Mr. R. Taylor.] In absolute ignorance of any — even the smallest link of those chains which bind life to matter, or that still more miraculous one, which connects mind with both, we can pursue our path only by the feeble light of analogy, and humbly hope that the Being, whose power and benevolence are unbounded, may enable us, in some further stage of our existence, to read another page in the history of his mighty works. Enough, however, and more than enough, may be gathered even from our imperfect acquaintance with matter, and some few of its laws, to prove the unbounded knowledge which must have preceded their organization.

Chapter IV. On the Account of the Creation, in the First Chapter of Genesis

A strange and singular argument has frequently been brought against the truth of the facts presented to us by Geology,—facts which every instructed person may confirm by the evidence of his senses. It has been asserted that they cannot be true; because, if admitted, they lead inevitably to the conclusion, that the earth has existed for an enormous period, extending, perhaps, over millions of years; whereas, it was supposed, from the history of the creation as delivered by Moses, that the earth was first created about six thousand years ago.

A different interpretation has been lately put upon that passage of the sacred writings; and, according to the highest authorities of the present time, it was not the intention of the writer of the book of Genesis to assign this date to the creation of our globe, but only to that of its most favored inhabitants.

Now, it is obvious that additional observations, and another advance in science, may at no distant period render necessary another interpretation of the Mosaic narrative; and this again, at a more remote time, may be superseded by one more in accordance with the existing knowledge of that day. And thus, the authority of Scripture will be gradually undermined by the weak though well-intentioned efforts of its friends in its support. For it is clear that when a work, translated by persons most highly instructed in its language, and seeking, in plainness and sincerity, to understand its true meaning, admits of such discordant interpretations, it can have little authority as a history of the past, or a guide to the future.

It is time, therefore, to examine this question by another light, and to point out to those who support what is called the literal interpretation of Scripture, the precipice to which their doctrines, if true, would inevitably lead; and to show, not by the glimmerings of elaborate criticism, but by the plainest principles of common sense, that there exists no such fatal collision between the words of Scripture and the facts of nature. And first, let us examine what must of necessity be the conclusion of any candid mind from the mass of evidence presented to it. Looking solely at the facts in which all capable of investigation agree—facts which it is needless to recite, they having been so fully and ably stated in the works of Mr. Lyell and Dr. Buckland, — we there see, and with no theoretic eye, the remains of animated beings, more and more differing from existing races, as we descend in the series of strata. Not merely are the petrified bones preserved, displaying marks of the insertion of every muscle necessary for the movement of the living animal, but in some cases we discover even the secretions of their organs, prepared either for nourishment or for defense. Almost every stratum we pause to examine, affords indubitable evidence of having, at some former period, existed for ages at the bottom of some lake or estuary, some inland sea, or some extensive ocean teeming with animal existence, or of having been the surface of a country covered with vegetation, which perished and was renewed at distant and successive periods.

Those, however, who, without the knowledge which enables them to form an opinion on the subject, feel any latent wish that this evidence should be overthrown, would do well to remember that geology also furnishes strong evidence in favor of the much more direct statement of Moses, as to the recent creation of man. And although we must ever feel a certain degree of caution in admitting negative evidence as conclusive; yet, in the present instance, the fact that the multitude of fossil bones which have been discovered, and which, when examined by persons duly qualified for the task, have been uniformly pronounced to be those of various tribes of animals, and not those of the human race, undoubtedly affords strong corroborative evidence in confirmation of the Mosaic account.

In truth, the mass of evidence which combines to prove the great antiquity of the earth itself, is so irresistible, and so unshaken by any opposing facts, that none but those who are alike incapable of observing the facts, and of appreciating the reasoning, can for a moment conceive the present state of its surface to have been the result of only six thousand years of existence. What, then, have those accomplished who have restricted the Mosaic account of the creation to that diminutive period, which is, as it were, but a span in the duration of the earth's existence, and who have imprudently rejected the testimony of the senses, when opposed to their philological criticisms? Undoubtedly, if they have succeeded in convincing either themselves or others, that one side of the question must be given up as untenable; those who are so convinced are bound to reject that which rests on testimony, not that which is supported by still existing facts. The very argument which Protestants have opposed to the doctrine of transubstantiation [the historian of the "Decline of the Roman Empire," carried the argument yet further:—"I still remember (he remarks) my solitary transport at the discovery of a philosophical argument against transubstantiation; that the text of Scripture which seems to indicate the real presence is attested only by a single sense — our sight; while the real presence itself is disproved by three of our senses — the sight, the touch, the taste." Gibbon's Memoirs of his Life, vol. i. p. 58], would, if their view of the case were correct, be equally irresistible against the book of Genesis.

But let us consider what would be the conclusion of every reasonable being in a parallel case. Let us imagine a manuscript written three thousand years ago, and professing to be a revelation from the Deity, in which it was stated that the color of the paper of the very book now in the reader's hands is black, and that the color of the ink in the characters which he is now reading is white:—with that reasonable doubt of his own individual faculties which would become the inquirer into the truth of a statement said to be derived from so high an origin, he would ask of all those around him, whether to their senses the paper appeared to be black and the ink to be white. If he found the senses of other individuals agree with his own, then he would undoubtedly pronounce the alleged revelation a forgery, and those who propounded it to be either deceived or deceivers. He would rightly impute the attempted deceit to moral turpitude, to gross ignorance, or to interested motives in the supporters of it; but he certainly would not commit the impiety of supposing the Deity to have wrought a miraculous change upon the senses of our whole species, and then to demand their belief in a fact directly opposed to those senses;—thus throwing doubt upon every conclusion of reason in regard to external objects, and amongst others, upon the very evidence by which the authenticity of that questionable manuscript was itself supported, and even upon the fact of its existence when before their eyes.

Thus, then, had those who attempt to show that the account of the creation, in the book of Genesis, is contradicted by the discoveries of modern science, succeeded, they would have destroyed the testimony of Moses—they would have uncanonized one portion of Scripture, and by implication have thrown doubt on the remainder. But minds which thus failed to trace out the necessary consequences of their own argument, were not likely to have laid very secure foundations for the basis on which it rested; and I shall presently prove that the contradiction they have imagined can have no real existence; and that whilst the testimony of Moses remains unimpeached, we may also be permitted to confide in the testimony of our senses.

Chapter V. Further View of the same Subject

Before entering on the main argument of the last Chapter, it may be remarked, that the plainest and most natural view of the language employed by the sacred historian is, that his expressions ought to be received by us in the sense in which they were understood by the people to whom he addressed himself. If, when speaking of the creation, instead of using the terms light and water, he had spoken of the former as a wave, and of the latter as the union of two invisible airs, he would assuredly have been perfectly unintelligible to his countrymen:— at the distance of above three thousand years his writings would just have begun to be comprehended; and possibly three thousand years hence those views may be as inapplicable to the then existing state of human knowledge as they would have been when the first chapter of Genesis was written. Those, however, who attempt to disprove the facts presented by observation, by placing them in opposition to revelation, have mistaken the very groundwork of the question. The revelation of Moses itself rests, and must necessarily rest, on testimony. Moses, the author of the oldest of the sacred books, lived about fifteen hundred years before the Christian era, or about three thousand three hundred years ago. The oldest manuscripts of the Pentateuch at present known, appear to have been written about 900 years ago. [Footnote.] These were copied from others of older date, and those again might probably, if their history were known, be traced up through a few transcripts to the original author; but no part of this history is revelation; it is testimony. Although the matter which the book contains was revealed to Moses, the fact that what we now receive as revelation is the same with that originally communicated revelation, is entirely dependent on testimony.

Mr. Horne, in the Introduction to the Critical Study of the Holy Scriptures, states, that the total number of Hebrew MSS. collated by Dr. Kennicott, for his critical edition of the Hebrew Bible, was about 630. In that work, Mr. Horne gives an account of ten of the most ancient of these MSS.: four of which contain the first chapter of Genesis, viz.:—

No. 4. Codex Caesense, in the Malatesta Library at Bologna, written about the end of the eleventh century.

No. 6. Codex Mediolanensis, written towards the close of the twelfth century. "The beginning of the book of Genesis, and the end of Leviticus and Deuteronomy, have been written by a later hand."

No. 8. Codex Parisiensis, 27, about the commencement of the twelfth century.

No. 10. Codex Parisiensis, 24, written at the beginning of the twelfth century.

In the same work is an account of six of the most ancient of four hundred and seventy-nine MSS. collated by M. De Rossi. Two of these contain the first chapter of Genesis; and the date of both is about the end of the eleventh or beginning of the twelfth century.

Of the Manuscripts of the Samaritan versions of the Pentateuch, cited in the same work, — one is the Codex 197, in the Ambrosian Library at Milan, which in the opinion of Dr. Kennicott is certainly not later than the tenth century.

Admitting, however, the full weight of that evidence above mentioned, corroborated as it is by the Samaritan version; nay, even supposing that we now possessed the identical autograph of the book of Genesis by the hand of its author, a most important question remains, — What means do we possess of translating it?

In similar cases we avail ourselves of the works of the immediate predecessors, and of the contemporaries of the writer; but here we are acquainted with no work of any predecessor,— with no writing of any contemporary; nor do we possess the works of any writers in the same language, even during several succeeding centuries, if we except some few of the sacred books. How, then, is it possible to satisfy our minds of the minute shades of meaning of words, perhaps employed popularly; or, if they were employed in a stricter and more philosophical sense, where are the contemporary philosophical writings from which their accurate interpretation may be gained?

The extreme difficulty of such an inquiry will be made apparent by imagining a parallel case. Let us suppose all writings in the English, and indeed in all other languages previous to the time of Shakespeare, to have been destroyed; — let us imagine one manuscript of his plays to remain, but not a vestige of the works of any of his contemporaries; and further, suppose the whole of the succeeding works of English literature to be annihilated nearly up to the present time. Under such circumstances, what would be our knowledge of Shakespeare? We should undoubtedly understand the general tenor and the plots of his plays. We should read the language of all his characters; and viewing it generally, we might even be said to understand it. But how many words connected with the customs, habits, and manners of the time must, under such circumstances, necessarily remain unknown to us! Still further, if any question arose, requiring for its solution a knowledge of the minute shades of meaning of words now long obsolete, or of terms supposed to be used in a strict or philosophical sense, how completely unsatisfactory must our conclusions remain! Such I conceive to be the view which common sense bids us take of the interpretation of the book of Genesis. The language of the Hebrews, in times long subsequent to the date of that book, may not have so far changed as to prevent us from understanding generally the history it narrates; but there appears to be no reasonable ground for venturing to pronounce with confidence on the minute shades of meaning of allied words, and on such foundations to support an argument opposed to the evidence of our senses.

I should have hesitated in offering these remarks respecting the right interpretation of the Mosaic account of the creation, had my argument depended on any acquaintance with the language in which the sacred volume is written, or on any refinements of criticism, had I possessed that knowledge; but in estimating its validity, or in supplying a more cogent argument, I entreat the reader to consider well the difficulties which it is necessary to meet.

1st. The Church of England, if we may judge by the writings of those placed in authority, has hitherto considered it to have been expressly stated in the book of Genesis, that the earth was created about six thousand years ago.

2dly. Those observers and philosophers who have spent their lives in the study of Geology, have arrived at the conclusion that there exists irresistible evidence, that the date of the earth's first formation is far anterior to the epoch supposed to be assigned to it by Moses; and it is now admitted by all competent persons, that the formation even of those strata which are nearest the surface must have occupied vast periods — probably millions of years — in arriving at their present state.

3dly. Many of the most distinguished members of the Church of England now distinctly and formally admit the fact of such a lengthened existence of the earth we inhabit. It is so stated in the eighth Bridgewater Treatise, a work written by the Professor of Geology in the University of Oxford — himself holding an office of dignity in that Church, and expressly appointed to write upon that subject, by the Archbishop of Canterbury, and the Bishop of London.

4thly. The Professor of Hebrew at the same University has proposed a new interpretation of those passages in the Book of Genesis, which were hitherto supposed to be adverse to the now admitted facts.

Such being the present state of the case;— it surely becomes a duty to require a very high degree of evidence, before we again claim authority for the opinion that the book of Genesis contains such a precise account of the work of the creation, that we may venture to appeal to it as a refutation of observed facts. The history of the past errors of our parent Church supplies us with a lesson of caution which ought not to be lost by its reformed successors. The fact that the venerable Galileo was compelled publicly to deny, on bended knee, a truth of which he had the most convincing demonstration, remains as a beacon to all after time, and ought not to be without its influence on the inquiring minds of the present day.

If the explanation offered by the Professor of Hebrew be admitted, those who adhere to it must still have some misgivings as to the probability that new discoveries in nature may give continual occasion for amended translations of various texts; whereas, should the view which has been advocated in this chapter be found correct, instead of fearing that the future progress of science may raise additional difficulties in the way of revealed religion, we are at once relieved from all doubt on that subject.

Chapter VI. Of the Desire of Immortality

The wish universally felt, and expressed in every variety of form, to remain in the memory of our fellow-creatures after our passage from the present scene, has rightly been adduced as an intimation of the desire of immortality, and has sometimes been explained as being founded on an instinctive belief that we are destined to be immortal by the Creator.

The hope of remaining embalmed in the fond recollection of those we held most dear in life, and even of being remembered by our more immediate descendants, has something in it nearly connected with self; but the wish for more extended reputation,—the desire that our name should pass in after times from mouth to mouth, cherished and admired by those whose applause is won by no personal recollections: or the still more fervent aspirations, that we may stamp indelibly on the age we live in some mark of our individual existence which shall form an epoch in the history of man: these hopes, these longings, receive no interpretation from the all-dominant principle of self; unless indeed we suppose the sentient principle of our nature not merely to exist, but also to be conscious of, and gratified by, the earthly immortality it had achieved. Yet the more distant and the higher the objects we pursue, the less is it possible to suppose the mind, so occupied on earth, can, in another stage of its existence, derive pleasure from such perceptions.

To support this opinion, it is only necessary to examine the states of mind in the various classes of the aspirants after fame.

Through every form of society, and through every rank of each, may be traced this universal passion. Examine the most highly civilized inhabitants of earth; search through it for the most cultivated and refined in taste; for the most sagacious in penetrating the passions of mankind, the most skillful in wielding them, or the most powerful in intellectual might. Taste, feeling, passion, ambition, genius, severed or combined, equally yield obedience to its sway, and present, under different appearances, the effects of its all-controlling power.

Look at the highest productions of the poet or the novelist. By connecting his story with the scenery, the traditions, or the history of his country, he may ensure for it a local interest, a domestic and transitory popularity; but it is that deeper penetration into the secrets of the human heart, which enables him to select from amongst the same materials, those feelings that are common to the race which have, as occasion called them forth, appeared, and will continue to reappear, as long as the same affections and passions shall continue to animate and agitate our frames.

From the examination of these its highest forms, we may gather some common principles, and be enabled to perceive that the love of fame is far different from that passion for vulgar applause with which it is too frequently confounded. We may learn, that the higher the intellectual powers devoted to the task, the more remote the period for which ambition delights to raise its far distant altar.

Chapter VII. "On Time"

Time and change are great, only with reference to the faculties of the beings which note them. The insect of an hour, fluttering, during its transient existence, in an atmosphere of perfume, would attribute unchanging duration to the beautiful flowers of the cistus, whose petals cover the dewy grass but a few hours after it has received the lifeless body of the gnat. These flowers, could they reflect, might contrast their transitory lives with the prolonged existence of their greener neighbors. The leaves themselves, counting their brief span by the lapse of a few moons, might regard as almost indefinitely extended the duration of the common parent of both leaf and flower. The lives of individual trees are lost in the continued destruction and renovation which take place in forest masses. Forests themselves, starved by the exhaustion of the soil, or consumed by fire, succeed each other in slow gradation. A forest of oaks waves its luxuriant branches over a spot which has been fertilized by the ashes of a forest of pines. These periods again merge into other and still longer cycles, during which the latest of a thousand forests sinks beneath the waves, from the gradual subsidence of its parent earth; or in which extensive inundations, by accumulating the silt of centuries, gradually convert the living trunks into their stony resemblances. Stratum upon stratum subsides in comminuted particles, and is accumulated in the depths of ocean, whence they again arise, consolidated by pressure and by heat, to form the continents and mountains of a new creation.

Such, in endless succession, is the history of the changes of the globe we dwell upon; and human observation, aided by human reason, has as yet discovered few signs of a beginning—no symptom of an end. Yet, in that more extended view which recognizes our planet as one amongst the attendants of a central luminary; that sun itself,—the soul, as it were, of vegetable and animal existence, but an insignificant individual among its congeners of the milky way:— when we remember that that cloud of light, gleaming with its myriad systems, is but an isolated nebula amongst a countless host of rivals, which the starry firmament surrounding us on all sides, presents in every varied form;— some as uncondensed masses of attenuated light;— some as having, in obedience to attractive forces, assumed a spherical figure; others, as if farther advanced in the history of their fate, enclosing a denser central nucleus surrounded by a more diluted light, spreading into such vast space, that the whole of our own nebula would be lost in it:—others there are, in which the apparently unformed and irregular mass of nebulous light is just curdling, as it were, into separate systems; whilst many present a congeries of distinct points of light, each, perhaps, the splendid luminary of a creation more glorious than our own;—when the birth, the progress, and the history of sidereal systems are considered, we require some other unit of time than even that comprehensive one which astronomy has unfolded to our view. Minute and almost infinitesimal as is the time which comprises the history of our race compared with that which records the history of our system, the space even of this latter period forms too limited a standard wherewith to measure the footmarks of eternity.

Chapter VIII. Argument from Laws intermitting—on the Nature of Miracles

The object of the present chapter is to show that it is more consistent with the attributes of the Deity to look upon miracles not as deviations from the laws assigned by the Almighty for the government of matter and of mind; but as the exact fulfillment of much more extensive laws than those we suppose to exist. In fact, if we were endued with acuter senses and higher reasoning faculties, they are the very points we should seek to observe, as the tests of any hypothesis we had been led to frame concerning the nature of those laws. Even with our present imperfect faculties we frequently arrive at the highest confirmation of our views of the laws of nature, by tracing their action under singular circumstances.

The mode by which I propose to arrive at these conclusions is, by again appealing to the judgment which each individual will himself form, when examining that piece of mere human mechanism, to which the argument so frequently compels me to advert. If he agrees with me, that the second of the two views presented to him exhibits a higher degree of knowledge, and a higher exertion of power, than the first, he must inevitably conclude, that the view here suggested of the nature of miracles, assigns a far higher degree of knowledge and power to the Deity.

Let the reader suppose himself placed before the calculating engine, and let him again observe and ascertain, by lengthened induction, the nature of the law it is computing. Let him imagine that he has seen the changes wrought on its face during the lapse of thousands of years, and that, without one solitary exception, he has found the engine register the series of square numbers. Suppose, now, the maker of that machine to say to the observer, "I will, by moving a certain mechanism, which is invisible to you, cause the engine to make one cube number instead of a square, and then to revert to its former course of square numbers;" the observer would be inclined to attribute to him a degree of power but little superior to that which was necessary to form the original engine.

But, let the same observer, after the same lapse of time — the same amount of uninterrupted experience of the uniformity of the law of square numbers, hear the maker of the engine say to him— "The next number which shall appear on those wheels, and which you expect to find a square number, shall not be so. When the machine was originally ordered to make these calculations, I impressed on it a law, which should coincide with that of square numbers in every case, except the one which is now about to appear; after which no future exception can ever occur, but the unvarying law of the squares shall be pursued until the machine itself perishes from decay."

Undoubtedly the observer would ascribe a greater degree of power to the artist who had thus willed that event which he foretells at the distance of ages before its arrival.

If the contriver of the engine then explain to him, that, by the very structure of it, he has power to order any number of such apparent deviations from its laws to occur at any future periods, however remote, and that each of these may be of a different kind; and, if he also inform him, that he gave it that structure in order to meet events, which he foresaw must happen at those respective periods, there can be no doubt that the observer would ascribe to the inventor far higher knowledge than if, when those events severally occurred, he were to intervene, and temporarily to alter the calculations of the machine.

If, besides this, the contriver were so far to explain the structure of the engine that the observer could himself, by some simple process, such as the mere moving of a bolt, call into action those apparent deviations whenever certain combinations were presented to his eye; if he were thus to impart a power of predicting such excepted cases, dependent on the will, though in other respects beyond the limits of the observer's power and knowledge, —such a structure would be admitted as evidence of a still more skilful contrivance.

The engine which, in a former chapter, I introduced to the reader, possesses these powers. It may be set, so as to obey any given law; and, at any periods, however remote, to make one or more seeming exceptions to that law. It is, however, to be observed, that the apparent law which the spectator arrived at, by an almost unlimited induction, is not the full expression of the law by which the machine acts; and that the excepted case is as absolutely and irresistibly the necessary consequence of its primitive adjustment, as is any individual calculation amongst the countless multitude which it may previously have produced.

When the construction of that engine was first attempted, I did not seek to give to it the power of making calculations so far beyond the reach of mathematical analysis as these appear to be: nor can I now foresee a probable period at which they may become practically available to human purposes. I had determined to invest the invention with a degree of generality which should include a wide range of mathematical power; and I was well aware that the mechanical generalizations I had organized contained within them much more than I had leisure to study, and some things which will probably remain unproductive to a far distant day.

Amongst those combinations which I was afterwards induced to examine, I observed the powers I have now recorded; and the reflections they produced in my own mind, impelled me to pursue them for a time. If the reader agrees with me in opinion, that these speculations lead to a more exalted view of the great Author of the universe than any we have hitherto possessed, he must also have arrived at the conclusion, that the study of the most abstract branch of practical mechanics, combined with that of the most abstruse portions of mathematical science, has no tendency to incapacitate the human mind from the perception of the evidences of natural religion; and that even those very sources themselves may furnish arguments which open views of the grandeur of creation perhaps more extensive than any which the sciences of observation or of physics have yet supplied.

It may not, perhaps, be without its use to suggest another illustration derived from the same quarter respecting the nature of miracles. It is known that mathematical laws are sometimes expressed by curves. The figure 1 represents a re-entering curve of four dimensions, whose law of formation is given in the note. [The equation y⁴ — 4 y² = — ax⁴ + bx³ + cx² + dx + e expresses several figures of an oval form, according to the nature of the roots of the equation, — ax⁴  + bx³ + cx² + dx + e = o.
If its two lesser roots become imaginary, the curves, figures 1, 2, 3 are produced.] A slight change in the nature of the constants makes it assume the form of fig. 2, which is still a continuous curve; but a further change of the constants causes it to have two ovals, quite disconnected from the larger portion; and, as the constants again alter, these ovals are reduced to points. In all four cases, every point in each branch of the curve obeys the same general law. The points, P and Q, though strictly invisible to the eye, are yet detected by mathematical analysis, and fulfill as precisely the original equation as any of the infinite number of other points, which constitute the rest of the curve. These points might be situated on the curve itself, and they are well known to mathematicians. It is to these singular points, which really fulfill the law of the curve, but which present to those who judge of them only by the organ of sight an apparent discontinuity, that I wish to call the attention, as offering an illustration of the doctrine here explained respecting miracles.

It has been remarked, in the beginning of the present chapter, that it is to the singular points—to those points of such infinitely rare occurrence in a curve—that we frequently have recourse, as the test of our theories, for explaining the phenomena of nature.

The existence of conical refraction in certain crystals, under peculiar circumstances, was predicted by Sir W. Hamilton; and, from an analytical investigation into the nature of the curve surface, which represents the form of the luminiferous wave within the crystal, he ascertained that it had four conoidal cusps, at each of which there were, consequently, an infinite number of tangent planes. The course of the refracted ray being determined by the tangent plane to the wave surface, it followed that a single ray within the crystal, transmitted in the direction of the line joining two opposite cusps, corresponded to an infinite number of refracted rays without, constituting a refracted cone.

A second case of conical refraction, predicted by Sir William Hamilton, depended on another mathematical fact—namely, that the wave surface is touched in an infinite number of points, constituting a small circle of contact, by a single plane parallel to one of the circular sections of the surface of elasticity.

Professor Lloyd undertook to make the very delicate experiments required for this most interesting investigation. Of its great importance he was fully aware, for he remarks—

"Here, then, are two singular and unexpected consequences of the undulatory theory, not only unsupported by any facts hitherto observed, but even opposed to all the analogies derived from experience. If confirmed by experiment, they would furnish new and almost convincing proofs of that theory; and, if disproved, on the other hand, it is evident that the theory must be abandoned or modified." [Trans. of Royal Irish Academy, Vol. XVII.]

On examining the first of these cases, experimentally, the fact of conical refraction was fully established. But a new result now presented itself: the rays of light thus conically refracted were found to be polarized; and it was observed, that "the angle between the planes of polarization of any two rays of the cone was half the angle between the planes, containing the rays themselves, and the axis."

This new law, thus approximately obtained by experiment, led the observer back to the theory; and, on a further examination, he detected in that theory the very law he had just discovered by observation.

The second case of conical refraction required experiments of a still more delicate nature. They were, however, made, and succeeded equally. The conically refracted ray was found to be polarized, according to the law which, in this instance, analysis had predicted; and, to complete the triumph of this union of theory and experiment, the measures in both cases, when made under proper circumstances, accorded with the theoretical conclusions, within such limits as might be fairly attributed to the unavoidable errors of observation.

It is worthy of remark, that, at first, two facts presented themselves, which seemed to be at variance with the theory. In the first place, the emergent rays formed a solid cone, instead of a conical surface; and, in the second place, the calculated angle, subtended by the sides of the cone, was only one half the observed angle. Both the facts were shown to depend upon the size of the aperture through which the light was admitted, and to arise from the rays which were inclined at small angles to the single theoretical direction. When the aperture was diminished, so as to be very small, (the case calculated by Sir William Hamilton,) then the cone of light became truly a conical surface, and the observed angle was the same as the calculated one. [Those who are acquainted with the history of astronomy, cannot fail to recall a parallel discrepancy between observation and calculation in the theory of gravity. It appeared to result from that law, that the motion of the moon's apogee was only one half of what observation proved it to be; and it is singular that Euler, D'Alembert, and Clairaut arrived, by different methods, at the same erroneous result; and the truth of the great law of gravity appeared for a time to be doubtful. Clairaut, however, having assumed that the law of gravity contained a term sensible only at small distances (such as that of the moon), re-calculated the question, and finding it necessary, in consequence of the existence of this term, to push his approximation further than he had done, arrived at the conclusion, that the co-efficient of the new term vanished; and also, that when the approximations were sufficiently pursued, the simple law of the inverse square of the distance accounted for the whole of the motions which observations had discovered.]

Chapter IX. On the permanent Impression of our Words and Actions on the Globe we inhabit

The principle of the equality of action and reaction, when traced through all its consequences, opens views which will appear to many persons most unexpected.

The pulsations of the air, once set in motion by the human voice, cease not to exist with the sounds to which they gave rise. Strong and audible as they may be in the immediate neighborhood of the speaker, and at the immediate moment of utterance, their quickly attenuated force soon becomes inaudible to human ears. The motions they have impressed on the particles of one portion of our atmosphere, are communicated to constantly increasing numbers, but the total quantity of motion measured in the same direction receives no addition. Each atom loses as much as it gives, and regains again from other atoms a portion of those motions which they in turn give up.

The waves of air thus raised, perambulate the earth and ocean's surface, and in less than twenty hours every atom of its atmosphere takes up the altered movement due to that infinitesimal portion of the primitive motion which has been conveyed to it through countless channels, and which must continue to influence its path throughout its future existence.

But these aerial pulses, unseen by the keenest eye, unheard by the acutest ear, unperceived by human senses, are yet demonstrated to exist by human reason; and, in some few and limited instances, by calling to our aid the most refined and comprehensive instrument of human thought, their courses are traced and their intensities are measured. If man enjoyed a larger command over mathematical analysis, his knowledge of these motions would be more extensive; but a being possessed of unbounded knowledge of that science, could trace every the minutest consequence of that primary impulse. Such a being, however far exalted above our race, would still be immeasurably below even our conception of infinite intelligence.

But supposing the original conditions of each atom of the earth's atmosphere, as well as all the extraneous causes acting on it to be given, and supposing also the interference of no new causes, such a being would be able clearly to trace its future but inevitable path, and he would distinctly foresee and might absolutely predict for any, even the remotest period of time, the circumstances and future history of every particle of that atmosphere.

Let us imagine a being, invested with such knowledge, to examine at a distant epoch the coincidence of the facts with those which his profound analysis had enabled him to predict. If any the slightest deviation existed, he would immediately read in its existence the action of a new cause; and, through the aid of the same analysis, tracing this discordance back to its source, he would become aware of the time of its commencement, and the point of space at which it originated.

Thus considered, what a strange chaos is this wide atmosphere we breathe! Every atom, impressed with good and with ill, retains at once the motions which philosophers and sages have imparted to it, mixed and combined in ten thousand ways with all that is worthless and base. The air itself is one vast library, on whose pages are for ever written all that man has ever said or woman whispered. There, in their mutable but unerring characters, mixed with the earliest, as well as with 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.

But if the air we breathe is the never-failing historian of the sentiments we have uttered, earth, air, and ocean, are the eternal witnesses of the acts we have done. The same principle of the equality of action and reaction applies to them: whatever movement is communicated to any of their particles, is transmitted to all around it, the share of each being diminished by their number, and depending jointly on the number and position of those acted upon by the original source of disturbance. The waves of air, although in many instances perceptible to the organs of hearing, are only rendered visible to the eye by peculiar contrivances; but those of water offer to the sense of sight the most beautiful illustration of transmitted motion. Every one who has thrown a pebble into the still waters of a sheltered pool, has seen the circles it has raised gradually expanding in size, and as uniformly diminishing in distinctness. He may have observed the reflection of those waves from the edges of the pool. He may have noticed also the perfect distinctness with which two, three, or more series of waves each pursues its own unimpeded course, when diverging from two, three, or more centers of disturbance. He may have seen, that in such cases the particles of water where the waves intersect each other, partake of the movements due to each series.

No motion impressed by natural causes, or by human agency, is ever obliterated. The ripple on the ocean's surface caused by a gentle breeze, or the still water which marks the more immediate track of a ponderous vessel gliding with scarcely expanded sails over its bosom, are equally indelible. The momentary waves raised by the passing breeze, apparently born but to die on the spot which saw their birth, leave behind them an endless progeny, which, reviving with diminished energy in other seas, visiting a thousand shores, reflected from each and perhaps again partially concentrated, will pursue their ceaseless course till ocean be itself annihilated.

The track of every canoe, of every vessel which 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.

The solid substance of the globe itself, whether we regard the minutest movement of the soft clay which receives its impression from the foot of animals, or the concussion arising from the fall of mountains rent by earthquakes, equally communicates and retains, through all its countless atoms, their apportioned shares of the motions so impressed.

Whilst the atmosphere we breathe is the ever-living witness of the sentiments we have uttered, the waters, and the more solid materials of the globe, bear equally enduring testimony of the acts we have committed.

If the Almighty stamped on the brow of the earliest murderer —the indelible and visible mark of his guilt,—he has also established laws by which every succeeding criminal is not less irrevocably chained to the testimony of his crime; for every atom of his mortal frame, through whatever changes its severed particles may migrate, will still retain, adhering to it through every combination, some movement derived from that very muscular effort, by which the crime itself was perpetrated.

The soul of the negro, whose fettered body surviving the living charnel-house of his infected prison, was thrown into the sea to lighten the ship, that his Christian master might escape the limited justice at length assigned by civilized man to crimes whose profit had long gilded their atrocity,—will need, at the last great day of human account, no living witness of his earthly agony. [The following extract is from a report by Captain Hayes to the Admiralty, of a representation made to him respecting one of these vessels in 1832. "The master having a large cargo of these human beings chained together, with more humanity than his fellows, permitted some of them to come on deck, but still chained together for the benefit of the air; when they immediately commenced jumping overboard, hand in hand, and drowning in couples; and, continued the person (relating the circumstance), 'without any cause whatever.' Now, these people were just brought from a situation between decks, and to which they knew they must return, where the scalding perspiration was running from one to the other, covered also with their own filth, and where it is no uncommon occurrence for women to be bringing forth children, and men dying at their side, with full in their view living and dead bodies chained together; and the living, in addition to all their other torments, laboring under the most famishing thirst (being in very few instances allowed more than a pint of water a day);—and, let it not be forgotten, that these unfortunate people had just been torn from their country, their families, their all! Men dragged from their wives, women from their husbands and children, girls from their mothers, and boys from their fathers; and yet in this man's eye (for heart and soul he could have had none) there was no cause whatever for jumping overboard and drowning. This, in truth, is a rough picture; but it is not highly colored. . .] When man and all his race shall have disappeared from the face of our planet, ask every particle of air still floating over the unpeopled earth, and it will record the cruel mandate of the tyrant. Interrogate every wave which breaks unimpeded on ten thousand desolate shores, and it will give evidence of the last gurgle of the waters which closed over the head of his dying victim: confront the murderer with every corporeal atom of his immolated slave, and in its still quivering movements he will read the prophet's denunciation of the prophet king. ["And Nathan said unto David — Thou art the man."]

Chapter X. On Hume's Argument against Miracles

Few arguments have excited greater attention, and produced more attempts at refutation, than the celebrated one of David Hume, respecting miracles; and it might be added, that more sophistry has been advanced against it, than its author employed in the whole of his writings.

It must be admitted that in the argument, as originally developed by its author, there exists some confusion between personal experience and that which is derived from testimony; and that there are several other points open to criticism and objection; but the main argument, divested of its less important adjuncts, never has, and never will be refuted. Dr. Johnson seems to have been of this opinion, as the following extract from his life by Boswell proves:—

"Talking of Dr. Johnson's unwillingness to believe extraordinary things, I ventured to say— "'Sir, you come near to Hume's argument against miracles — That it is more probable witnesses should lie, or be mistaken, than that they should happen.'

"Johnson.— Why, Sir, Hume, taking the proposition simply, is right. But the Christian revelation is not proved by miracles alone, but as connected with prophecies, and with the doctrines in confirmation of which miracles were wrought.'" [Boswell's Life of Johnson. Oxford, 1826. vol. iii. p. 169.]

Hume contends that a miracle is a violation of the laws of nature; and as a firm and unalterable experience has established these laws, the proof against a miracle from the very nature of the fact, is as entire as any argument from experience can possibly be imagined.

"The plain consequence is (and it is a general maxim worthy of our attention), that no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavors to establish: and even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force which remains after deducting the inferior." [Hume's Essays, Edinburgh, 1817, vol. ii. p. 117.]

The word miraculous employed in this passage is evidently equivalent to improbable, although the improbability is of a very high degree.

The condition, therefore, which, it is asserted by the argument of Hume, must be fulfilled with regard to the testimony, is that the improbability of its falsehood must be greater than the improbability of the occurrence of the fact.

This is a condition which, when the terms in which it is expressed are understood, immediately commands our assent. It is in the subsequent stage of the reasoning that the fallacy is introduced. Hume asserts, that this condition cannot be fulfilled by the evidence of any number of witnesses, because our experience of the truth of human testimony is not uniform and without any exceptions; whereas, our experience of the course of nature, or our experience against miracles, is uniform and uninterrupted.

The only sound way of trying the validity of this assertion is to measure the numerical value of the two improbabilities, one of which it is admitted must be greater than the other; and to ascertain whether, by making any hypothesis respecting the veracity of each witness, it is possible to fulfill that condition by any finite number of such witnesses.

Hume appears to have been but very slightly acquainted with the doctrine of probabilities, and, indeed, at the period when he wrote, the details by which the conclusions he had arrived at could be proved or refuted were yet to be examined and arranged. It is, however, remarkable that the opinion he maintained respecting our knowledge of causation is one which eminently brings the whole question within the province of the calculus of probabilities. In fact, its solution can only be completely understood by those who are acquainted with that most difficult branch of science. By those who are not so prepared, certain calculations, which will be found more fully developed in the Note ([Appendix] E), must be taken for granted; and all that can be attempted will be, to convey to them a general outline of the nature of the principles on which these enquiries depend.

A miracle is, according to Hume, an event which has never happened within the experience of the whole human race. Now, the improbability of the future happening of such an occurrence may be calculated according to two different views.

We may conceive an urn, containing only black and white balls, from which m black balls have been successively drawn and replaced, one by one; and we may calculate the probability of the appearance of a white ball at the next drawing. This would be analogous to the case of one human being raised from the dead after m instances to the contrary.

Looking, in another point of view, at a miracle, we may imagine an urn to contain a very large number of tickets, on each of which is written one of the series of natural numbers. These being thoroughly mixed together, a single ticket is drawn: the prediction of the particular number inscribed on the ticket about to be drawn may be assimilated to the occurrence of a miracle.

According to either of these views, the probability of the occurrence of such an event by mere accident may be calculated. Now, the reply to Hume's argument is this: Admitting at once the essential point, viz. that the improbability of the concurrence of the witnesses in falsehood must be greater than the improbability of the miracle, it may be denied that this does not take place. Hume has asserted that, in order to prove a miracle, a certain improbability must be greater than another; and he has also asserted that this never can take place.

Now, as each improbability can be truly measured by number, the only way to refute Hume's argument is by examining the magnitude of these numbers. This examination depends on known and admitted principles, for which the reader, who is prepared by previous study, may refer to the work of Laplace, Théorie Analytique des Probabilités; Poisson, Recherches sur la Probabilité des Jugements, 1837; or he may consult the article Probabilities, by Mr. De Morgan, in the Encyclopaedia Metropolitana, in which he will find this subject examined.

One of the most important principles on which the question rests, is the concurrence of the testimony of independent witnesses. This principle has been stated by Campbell, and has been employed by the Archbishop of Dublin [Elements of Rhetoric, by R. Whately, D.D. p. 57, 1832], and also by Dr. Chalmers. [Evidence of Christian Revelation, vol. i. p. 129.] It requires however to be combined with another principle, in order to obtain the numerical values of the quantities spoken of in the argument. The following example may be sufficient for a popular illustration.

Let us suppose that there are witnesses who will speak the truth, and who are not themselves deceived in ninety-nine cases out of a hundred. Now, let us examine what is the probability of the falsehood of a statement about to be made by two such persons absolutely unknown to and unconnected with each other.

Since the order in which independent witnesses give their testimony does not affect their credit, we may suppose that, in a given number of statements, both witnesses tell the truth in the ninety-nine first cases, and the falsehood in the hundredth.

Then the first time the second witness B testifies, he will agree with the testimony of the first witness A, in the ninety-nine first cases, and differ from him in the hundredth. Similarly, in the second testimony of B, he will again agree with A in ninety-nine cases, and differ in the hundredth, and so on for ninety-nine times; so that, after A has testified a hundred, and B ninety-nine times, we shall have

99 X 99 cases in which both agree,
99 cases in which they differ, A being wrong.

Now, in the hundredth case in which B testifies, he is wrong; and, if we combine this with the testimony of A, we have ninety-nine cases in which A will be right and B wrong; and one case only in which both A and B will agree in error. The whole number of cases, which amounts to ten thousand, may be thus divided:—

99 x 99 =9801 cases in which A and B agree in truth,
1 x 99 = 99 cases in which B is true and A is false,
99 x 1 = 99 cases in which A is true and B false,
1 x 1 = 1 cases in which both A and B agree in a falsehood.
-------
10,000 cases

As there is only one case in ten thousand in which two such independent witnesses can agree in error, the probability of their future testimony being false is

1/10,000 or 1/(100)²

The reader will already perceive how great a reliance is due to the future concurring testimony of two independent witnesses of tolerably good character and understanding. It appears that, previously to the testimony, the chance of one such witness being in error is 1/(100)¹; that of two concurring in the same error  is 1/(100)² and if the same reasoning be applied to three independent witnesses, it will be found that the probability of their agreeing in error is 1/(100)³; or that the odds are 999,999 to 1 against the agreement.

Pursuing the same reasoning, the probability of the falsehood of a fact which six such independent witnesses attest is, previously to the testimony, 1/(100)⁶ or it is, in round numbers,

1,000,000,000,000 to 1 against the falsehood of their testimony.

The improbability of the miracle of a dead man being restored, is, on the principles stated by Hume, 1/(20 (100)⁵) or it is —

200,000,000.000 to 1 against its occurrence.

It follows, then, that the chances of accidental or other independent concurrence of only six such independent witnesses, is already five times as great as the improbability against the miracle of a dead man's being restored to life, deduced from Hume's method of estimating its probability solely from experience.

This illustration shows the great accumulation of probability arising from the concurrence of independent witnesses: we must however combine this principle with another, before we can arrive at the real numerical value of the improbabilities referred to in the argument.

The calculation of the numerical values of these improbabilities I have given in Note (E.)

From this it results that, provided we assume that independent witnesses can be found of whose testimony it can be stated that it is more probable that it is true than that it is false, we can always assign a number of witnesses which will, according to Hume's argument, prove the truth of a miracle.

Chapter XI. Reflections on the Inquiry in the preceding

Those who have too hastily lent the credit of their authority to support the prejudice, that the pursuit of mathematical investigations renders the mind unapt for the perception of the truths of religion, must now be compelled to admit that they have endeavored to discredit a science, which alone can furnish an exact refutation of one of the most celebrated arguments against revelation. Those on the other hand, who, without the knowledge which qualifies men to form opinions on such subjects, have obtruded their dogmatism on  the world, though they may fail to learn the lessons of caution and modesty, may yet be disposed to abstain from scattering their misdirected darts, lest they should injure one of the best allies of the faith which they profess. Whilst those, who in humble diffidence of their own powers, rely on the opinions of others, and are believers from feeling rather than from reason, may learn from experience what they might fail to ascertain by inquiry, "that no TRUTH in any department of knowledge can ever be in contradiction to any other TRUTH;"— and they may after many such instances perceive, that it is no narrow evidence which has convinced the most enlightened men, that unlimited discussion is the fatal enemy of error—the most certain supporter of truth.

In the course of this inquiry, the gradually decreasing value of human testimony, when transmitted from witness to witness, has been alluded to, and its bearing on the future reception of revelation must have been foreseen by some, and may, when pointed out, be feared by others, of my readers. It may be apprehended, that after thousands of years, the long transmitted testimony of revealed truth, may not have sufficient force to convince the inquirers of that distant time. That such a cause is in constant action must be fully admitted, for every statement from man to his fellow man is liable to error from two sources. The witness may be deceived, or he may himself be a deceiver; and however extensive his knowledge, or however high his general character for veracity, whilst the possibility of a failure in either direction remains, repeated transmission through a series of such witnesses, will ultimately reduce to insignificance any statement in itself highly improbable. To suppose human beings incapabable of being deceived and incapable of deception, is to assign to them attributes which we know they do not possess, and which we can scarcely assign to responsible and created beings.

Let us examine then what reply reason and science can make to such forebodings. If a transmitted revelation contain within its pages a prophecy of events, dark and unintelligible in itself, and therefore unfit to cause its own fulfillment; and if from time to time facts occur, explaining instantly by no circuitous or lengthened process, but clearly and explicitly, the mystic words; if the explanation of that, which till then was dark and mysterious even to the learned and reflecting, flashes with spontaneous conviction on the minds of multitudes, who now discover for the first time the events to have been clearly predicted;—then a revelation, however faint from the lapse of time, revives with renewed energy, and claims its reception with a force almost equal to that which it demanded from those to whom it was originally delivered.

If, on the other hand, an inspired writer had given an account of a former state of our planet, different wholly from its present condition, yet so distinct and minute, that if there were placed before us a map of the islands, continents, and rivers, with the plants and the animals of that ancient world, we should instantly recognize the coincidence with the prophet's description, and if it were impossible, from the state of knowledge when the revelation was delivered, that the writer could have been acquainted even with the relics of that former world he so well described, then it must be admitted, that we should have strong and irresistible evidence of his veracity.

Let us suppose an inspired writer to describe a former constitution of our globe, in which a vast continent occupied the position now filled by the Pacific Ocean; that a great river with three outlets poured its waters towards the south; that these streams and their banks were peopled with animals differing as much from all then known races, as the plesiosaurus and the pterodactyl do from those which now inhabit our globe, and that he had described, not with anatomical precision but in popular language, the number of bones in the several parts of their frames, their habits, and food, as well as the plants which flourished on the banks of those rivers.

Thousands, or perhaps hundreds of thousands of years hence, we may conceive an island rising from the same ocean, and the geologists of that distant era tracing in its then elevated strata, by their fossil shells of mingled fresh water and marine origin, the estuaries of the streams from which the strata were deposited; discovering by the depth of their branches and their magnitude, that the river must have been formed from the drainage of some great continent; and finally, that aided by the comparative anatomists and botanists of that day, they should reconstruct from the fossil bones embedded in the strata, the very animals and plants described by the prophet, and ascertain even their habits and their food. If similar discoveries and reconstructions of animals previously unknown, have been made in our own times, almost in the infancy of these sciences, what advances may we not expect with the progress of time?

What has been stated by way of illustration as resulting from some branches of natural science, is equally applicable under different circumstances to many others. Nor does this view present any thing irreconcilable with the wisdom and benevolence of the Creator. In the early stages of the world, before man had acquired knowledge to read the book of nature ever open to his view, direct revelation might be as necessary for his belief in a Deity, as for his moral government; and this might from time to time be repeated. When civilization and science had fixed their abode amongst mankind, and when observations and reason had enabled man to penetrate some little way into the mysteries of nature, his conviction of the existence of a first great cause would gradually acquire additional strength from the use of his own faculties, and when accumulating proofs had firmly established this great step, the recurrence of revelation might be less necessary for his welfare.

The ancient revelation would however necessarily lose a portion of its weight from its continued transmission, and thus by slow but inevitable steps, tend in the lapse of ages to extinction. It is possible, however, that that very revelation may contain within its pages the verification of its own truth, and that the advancement of man in the knowledge of the structure of the works of his Creator, might furnish continually increasing proofs of its authenticity; and that thus by the due employment of our faculties, we might not merely redeem revelation from the ravages of time, but give to it a degree of force strengthening with every accession to our knowledge, and ultimately forcing our understandings to assent to it even with a conviction great as that which had compelled the belief of those to whom it was originally delivered.

It is not for the finite faculties of man to pronounce what has been the course chosen for his happiness by an all-wise Creator, but it may be permitted to him to meet the difficulty which necessarily arises from the fallible and fading nature of human testimony by pointing out one possible course in which, by the exertion of the highest faculties with which we have been blessed, we may make a nearer approach to the knowledge of the will of our Creator.

Chapter XII. On the Nature of a Superintending Providence

Some of the readers of the former edition of this work, whilst they have admitted the exalted view of the Creator, which arises from considering his will as the development of laws of unbounded generality, have expressed regret that those views appeared to them, in some measure, to imply that we are less immediately the objects of his protecting care, — that he seems thus less constantly and less directly, our watchful Guardian and Protector.

This difficulty ultimately resolves itself into that of the reconciling foreknowledge with freewill. Without, however, entering at present on that abstruse discussion, there are some reflections which may at least afford a partial reply to the objection stated above.

In reverting to the first origin of our knowledge of the motives and feelings of those around us, we must necessarily look into our own hearts. When we exercise that feeling which is called kindness or benevolence, we are conscious that we make some exertion, or some sacrifice, to add to the happiness of another person, without ourselves expecting to derive any advantage in return. When we observe other persons making similar exertions or sacrifices, and when we can discover no possibility of their deriving advantage from them, except that interior feeling of happiness which always arises from the exercise of such good offices, we conclude that they are actuated by the same feeling of kindness or benevolence. In the first case we are conscious of the existence of that feeling within our own bosoms; in the second we infer that it exists in others, from the similarity of human actions under the same apparent circumstances.

The reasoning which leads us to ascribe the attribute of benevolence to the Creator, is of precisely the same kind, although the infinity of this, as of his other attributes, can bear no comparison to the finite extent of those of other beings. Since however it is by such reasoning only that we attain the knowledge of them, so, if there arise a question about the comparative amount of any of those attributes when exerted in one way rather than in another, we must apply to such cases the same reasoning.

The inference from which the objection has arisen, is that the superintendence of Providence is remote, not immediate; the answer that I shall endeavor to make to it is, that the value of benevolence is not diminished by the distance of time at which its exertion arises.

In order to examine this question of immediate or remote superintendence, let us imagine a case occurring amongst our fellow-creatures, and let us endeavor to ascertain the conclusion we should form with regard to it.

Let us suppose the wealthy resident near a small village, returning from the neighboring town, to have observed from a distance that the bridge across a rapid stream has been carried away by a flood, and that the blind postman who makes his daily journey along the causeway to the neighboring town, unconscious of what has happened, is just approaching the torrent. Setting spurs to his horse, he dashes forward, and clearing the broken bridge, has just time to alight and catch the postman on the verge of destruction. Such conduct is an example of kindness or benevolence, and would receive the praise it justly merits.

Let us now suppose another village,— another stream,—and another postman starting, like the former one, at an early hour, before the villagers are abroad. We will imagine too the wealthy resident near this other village to have been out on the preceding day amongst the distant mountains, from which the torrent which passes the village is fed; and that observing the quantity of rain which has fallen in those parts, he foresees that the resulting flood will in all probability destroy the bridge across that stream; and knowing from his experience that several hours must elapse before the rain which falls in the mountains produces its full effect on the stream in his neighborhood, he sends off one of his attendants in time to reach the broken bridge just before the moment when the stream would be crossed by the unsuspecting traveller, who is thus saved from death.

Undoubtedly this is benevolence. And although we may not infer, that the benevolence in the latter, was greater than in the former case, yet it cannot for a moment be maintained to be less. In fact, the first case was one of benevolence excited by feeling; in the second it was benevolence called forth by reflection, and aided and reduced to action by reasoning founded upon knowledge. The sportsman in the mountains who thus reasoned, would not have been less active had he been on the spot at the dangerous moment. But he who by his personal exertion preserved the blind man, although of an equally benevolent heart, might not have possessed the knowledge available for the safety of the postman, had he been at a distance from the spot on which his effort was successful.

Two inferences are connected with this imagined case.

1st, That the benevolence which organizes beforehand contrivances for the advantage or security of its objects, is at least as high as that which acts from the impulse of immediate feeling.

2d, That the benevolence which is guided by knowledge, even though, as a feeling, it may not be superior in intensity, is often of far more value to those who are its subjects.

Such are the decisions we should arrive at respecting human feelings and human knowledge; and applying the same principles instead of looking with any feelings of doubt or apprehension at the distance of time at which it may have pleased the Creator to have organized our protection from danger, we ought, when once convinced of his benevolence, to discover in that very distance, proof of his higher knowledge and higher power, and with every addition to those attributes, to feel ourselves under still more potent and unintermitted watchfulness.

Chapter XIII. A priori Argument in favor of the Occurrence of Miracles

In the present chapter it is proposed to prove, that —

It is more probable that any law, at the knowledge of which we have arrived by observation, shall be subject to one of those violations which, according to Hume's definition, constitutes a miracle, than that it should not be so subjected.

To show this, we may be allowed again to revert to the Calculating Engine: and to assume that it is possible to set the machine, so that it shall calculate any algebraic law whatsoever: and also possible so to arrange it, that at any periods, however remote, the first law shall be interrupted for one or more times, and be superseded by any other law; after which the original law shall again be produced, and no other deviation shall ever take place.

Now, as all laws, which appear to us regular and uniform in their course, and to be subject to no exception, can be calculated by the engine: and as each of these laws may also be calculated by the same machine, subject to any assigned interruption, at distinct and definite periods; each simple law may be interrupted at any point by the temporary action of a portion of any one of all the other simple laws: it follows, that the class of laws subject to Interruption is far more extensive than that of laws which are uninterrupted. It is, in fact, infinitely more numerous. Therefore, the probability of any law with which we have become acquainted by observation being part of a much more extensive law, and of its having, to use mathematical language, singular points or discontinuous functions contained within it, is very large.

Perhaps it may be objected, that the laws calculated by such an engine as I have referred to are not laws of nature; and that any deviation from laws produced by human mechanism does not come within Hume's definition of miracles. To this it may be answered, that a law of nature has been defined by Hume to rest upon experience, or repeated observation, just as the truth of testimony does. Now, the law produced by the engine may be arrived at by precisely the same means—namely, repeated observation.

It may, however, be desirable to explain further the nature of the evidence, on which the fact, that the engine possesses those powers, rests.

When the Calculating Engine has been set to compute the successive terms of any given law, which the observer is told will have an apparent exception (at, for example, the ten million and twenty-third term,) the observer is directed to note down the commencement of its computations; and, by comparing these results with his own independent calculations of the same law, he may verify the accuracy of the engine as far as he chooses. It may then be demonstrated to him, by the very structure of the machine, that if its motion were continued, it would, necessarily, at the end of a very long time, arrive at the ten-millionth term of the law assigned to it; and that, by an equal necessity, it would have passed through all the intermediate terms. [This can be done in a few minutes.] The inquirer is now desired to turn on the wheels with his own hand, until they are precisely in the same situation as they would have been had the engine itself gone on continuously, to the ten-millionth term. The machine is again put in motion, and the observer again finds that each successive term it calculates fulfills the original law. But, after passing twenty-two terms, he now observes one term which does not fulfill the original law, but which does coincide with the predicted exception.

The continued movement now again produces terms according with the first law, and the observer may continue to verify them as long as he wishes. It may then be demonstrated to him, by the very structure of the machine, that, if its motion were continued, it would be impossible that any other deviation from the apparent law could ever occur at any future time.

Such is the evidence to the observer; and, if the superintendent of the engine were, at his request, to make it calculate a great variety of different laws, each interrupted by special and remote exceptions, he would have ample ground to believe in the assertion of its director, that he could so arrange the engine that any law, however complicated, might be calculated to any assigned extent, and then there should arise one apparent exception; after which the original law should continue uninterrupted for ever.

Let us now consider the miracle alluded to by Hume—the restoration of a dead man to life. According to the definition of that author, our belief that such a fact is contrary to the laws of nature, arises from our uniform experience against it. Our personal experience is small: we must therefore have recourse to testimony; and from that we learn, that the dead are never restored to life; and, consequently, we have the uniform experience of all mankind since the creation, against one assigned instance of a dead man being so restored. Let us now find the numerical amount of this evidence. Assuming the origin of the human race to have been about six thousand years ago, and taking thirty years as the duration of a generation, we have —

6000/30 = 200 generations

And allowing that the average population of the earth has been a thousand millions, we find that there have been born and have died since the creation,

200 x 1,000,000,000 = 2,000,000,000 individuals

Such, then, according to Hume, are the odds against the truth of the miracle: that is to say, it is found from experience, that it is about two hundred thousand millions to one against a dead man having been restored to life.

Let us now compare this with a parallel case in the calculations of the engine; let us suppose the number above stated to be a hundred million times as great, or that the truth of the miracles is opposed by a number of instances, expressed by twenty places of figures.

The engine may be set to count the natural numbers — 1, 2, 3, 4, &c.; and it shall continue to fulfill that law, not merely for the number of times just mentioned, for that number is quite insignificant among the vast periods it involves; but the natural numbers shall follow in continual succession, until they have reached an amount which requires for its expression above a hundred million places of figures. If every letter in the volume now before the reader's eyes 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 then have had in favor of the truth of the law of natural numbers. The widest range of all the cycles of astronomy and geology combined, sink into insignificance before such a period. Yet, shall the engine, true to the prediction of its director, after the lapse of myriads of ages, fulfill 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? It would have had, according to Hume, the evidence of all experience against it, with a force myriads of times more strong than that against any miracle.

Now, let the reader, who has fully entered into the nature of the argument, ask himself this question:—Does he believe that such an engine has really been contrived, and what reasonable grounds has he for that belief?

The testimony of any single witness is small against such odds; besides, the witness may deceive himself. Whether he speaks truly, will be estimated by his moral character—whether he deceives himself, will be estimated by his intellectual character. The probability that such an engine has been contrived, will, however, receive great addition, when it is remarked, that mathematical—and, especially, geometrical evidence is, of all others, that in which the fewest mistakes arise, and in which they are most readily discovered; and when it is added, that the fact of the invention of such an engine rests on precisely the same species of evidence as the propositions of Euclid, and may be deduced from the drawings with all the force of demonstration. Whether such an engine could be actually made in the present state of mechanical art, is a question of quite a different order: it must rest upon the opinions of those who have had extensive experience in that art. The author has not the slightest hesitation in stating his opinion to be, that it is fully within those limits.

This, however, is a question foreign to the nature of the argument, which might have been stated in a more abstract manner, without any reference to such an engine. As, however, the argument really arose from that machine, and as visible forms make a much deeper impression on the mind than any abstract reasonings, it has been stated in conjunction with that subject.

Chapter XIV. Thoughts on the Nature of Future Punishments

Who has not felt the painful memory of departed folly? Who has not at times found crowding on his recollection, thoughts, feelings, scenes, by all perhaps but himself forgotten, which force themselves involuntarily on his attention? Who has not reproached himself with the bitterest regret at the follies he has thought, or said, or acted? Time brings no alleviation to these periods of morbid memory: the weaknesses of our youthful days, as well as those of later life, come equally unbidden and unarranged, to mock our attention and claim their condemnation from our severer judgment.

It is remarkable that those whom the world least accuses, accuse themselves the most; and that a foolish speech, which at the time of its utterance was unobserved as such by all who heard it, shall yet remain fixed in the memory of him who pronounced it, with a tenacity which he vainly seeks to communicate to more agreeable subjects of reflection. It is also remarkable that whilst our own foibles, or our imagined exposure of them to others, furnish the most frequent subject of almost nightly regret, yet we rarely call to recollection our acts of consideration for the feelings of others, or those of kindness and benevolence. These are not the familiar friends of our memory, ready at all times to enter the domicile of mind its welcome but unbidden guests. When they appear, they are usually summoned at the command of reason, to meet some expected ingratitude, or when the mind retires within its council chamber to nerve itself for the endurance or the resistance of injustice.

If such be the pain, the penalty of thoughtless folly, who shall describe the punishment of real guilt? Make but the offender better, and he is already severely punished. Memory, that treacherous friend but faithful monitor, recalls the existence of the past, to a mind now imbued with finer feelings, with sterner notions of justice than when it enacted the deeds thus punished by their recollection.

If additional knowledge be given to us, the consequences of many of our actions appear in a very altered light. We become acquainted with many evils they have produced, which, although quite unintentional on our part, are yet subjects of painful regret. But this unavailing regret is mixed with another feeling far more distressing. We reproach ourselves with not having sufficiently employed the faculties we possessed in acquiring that knowledge, which, if we had attained it, would have prevented us from committing acts we now discover to have been injurious to those we best loved.

On the other hand, the good which such increased knowledge enables us to discover that we have unintentionally done, fails to produce the satisfaction always arising from a virtuous motive; and it is accompanied by the regret that, by a sufficient cultivation of our faculties, we might have enjoyed a still higher gratification, by a more efficient service to our fellow-creatures.

Thus, on whichsoever side we look at the question, knowledge alone is advantageous to virtue; and if additional knowledge alone were given in a future life, it would cause the best of us to regret the errors of the present.

Let us now consider the consequences of a higher tone of moral feeling—of a perception of excellencies of character in others, hitherto unappreciated.

Without the torment arising from additional knowledge, we may, in such circumstances, perceive, that the pain we have inflicted for imagined offences was quite beyond their real deserts; and may feel that the justice we have done to others, has been quite disproportioned to the sacrifices they have made to serve us.

If, without any addition to our intellectual faculties, increased perfection were given to our bodily senses, the same result would ensue. Wollaston has shown, that there are sounds of such a nature, that they can be heard by some individuals, but are inaudible to others,—a circumstance which may arise either from the incapacity of the parts of the ear to vibrate in the same time as those producing the sound, or from the force of the sounding body being insufficient to communicate motion through the air to those portions of the ear whose movement is required to produce the sensation of hearing.

If we imagine the soul in an after stage of our existence, to be connected with a bodily organ of hearing so sensitive, as to vibrate with motions of the air, even of infinitesimal force, and if it be still within the precincts of its ancient abode, all the accumulated words pronounced from the creation of mankind, will fall at once upon that ear. Imagine, in addition, a power of directing the attention of that organ entirely to any one class of those vibrations: then will the apparent confusion vanish at once; and the punished offender may hear still vibrating on his ear the very words uttered, perhaps, thousands of centuries before, which at once caused and registered his own condemnation.

It seems, then, that either with improved faculties or with increased knowledge, we could scarcely look back with any satisfaction on our past lives;—that, to the major part of our race, oblivion would be the greatest boon. But if, in a future state, we could turn from the contemplation of our own imperfections, and with increased powers apply our minds to the discovery of nature's laws, and to the invention of new methods by which our faculties might be aided in that research, pleasure the most unalloyed would await us at every stage of our progress. Unclogged by the dull corporeal load of matter which tyrannizes even over our most intellectual moments, and chains the ardent spirit to its unkindred clay, we should advance in the pursuit, stimulated instead of wearied by our past exertions, and encountering each new difficulty in the inquiry, with the accumulated power derived from the experience of the past, and the irresistible energy resulting from the confidence of ultimate success.

Whether, then, we regard our future prospects as connected with a far higher acuteness of our present senses—or, as purified by more exalted moral feelings—or, as guided by intellectual power surpassing all we contemplate upon earth, we equally arrive at the conclusion, that the mere employment of such enlarged faculties, in surveying our past existence, will be an ample punishment for all our errors; whilst, on the other hand, if that Being who assigned to us those faculties, should turn their application from the survey of the past, to the inquiry into the present and to the search into the future, the most enduring happiness would arise from the most inexhaustible source.

Chapter XV. Reflections on Free Will

The great question of the incompatibility of one of the attributes of the Creator—that of fore-knowledge, with the existence of the free exercise of their will in the beings he has created,—has long baffled human comprehension; nor is it the object of this chapter to enter upon that difficult question.

As, however, some of the properties of the Calculating Engine seem, although but very remotely, to bear on a similar question, with respect to finite beings, it may, perhaps, not be entirely useless to state them.

It has already been observed, that it is possible so to adjust the engine, that it shall change the law it is calculating into another law, at any distant period which may be assigned.

Now, by a similar adjustment, this change may be made to take place at a time not foreseen by the person employing the engine. For example: when calculating a table of squares, it may be made to change into a table of cubes, the first time the square number ends in the figures —

269696;

an event which only occurs at the 99736th calculation; and whether that fact is known to the person who adjusts the machine or not, is immaterial to the result. But the very condition on which the change depends, may be impossible. Thus, the change of the law from that of squares to that of cubes may be made to take place the first time the square number ends in 7.

But it is known, that no square number can end in a 7; consequently the event, on the happening of which the change is determined, can itself never take place. Yet, the engine retains impressed on it a law, which would be called into action if the event on which it depends could occur in the course of the law it is calculating.

Nay, further, if the observer of the engine is informed, that at certain times he can move the last figure the engine has calculated, and change it into any other, in consequence of which it becomes possible that some future term may end in 7; then, after he has so changed the last figure, whenever that terminal figure arrives, all future numbers calculated by the machine will follow the law of the cubes.

These contingent changes may be limited to single exceptions, and the arrival of such an exception may be made contingent on a change which is only possible at certain rare periods. For example: the engine may be set to calculate square numbers, and after a certain number of calculations — ten million and fifty-three, for instance, it shall be possible to add unity to a wheel in another part of the engine, which in every other case is immovable. This fact being communicated to the observer, he may either make that addition or refrain from it: if he refrain, the law of the squares will continue for ever; if he make the addition, one single cube will be substituted for that square number, which ought to occur ten million and five terms beyond the point at which he made the addition; and after that no future addition will ever become possible, and no deviation from the law of the squares ever can occur.

Conclusion

Reader, I have now fulfilled the task I undertook. Laboring under that imputed mental incapacity which the science I cultivate has been stated to produce, I have brought from the recesses of that science the reasonings and illustrations by which I have endeavored faintly to embody the human conception of the Almighty mind. It is for you to determine whether the trains of thought I have excited have lowered or exalted your previous notions of the power and the knowledge of the Creator.

That prejudice which I have endeavored to expose is not a merely speculative opinion, it is a practical evil; and those whose writings have been supposed to give support to it, will, I am sure, feel grieved when they learn that it is used by the ignorant and the designing, for the injury of the virtuous and the instructed; that it is employed as a firebrand, to disturb the relations of social life. They will also, if the arguments I have used have the same effect on their minds which they have had upon my own, lament still more deeply that they should have contributed, in any degree, to throw discredit on that species of knowledge which is now found to supply some of the strongest arguments in favor of religion. I will, however, hope that the opinions I have combated are not shared or even countenanced by the higher authorities of our Protestant Church; and I cannot better conclude this Fragment, than by recalling to the reader the words of one, whose power of reasoning, and whose love of truth, add dignity to the high station he so deservedly fills:—

"Lastly, As we must not dare to withhold or disguise revealed religious truth, so, we must dread the progress of no other truth. We must not imitate the bigoted Romanists who imprisoned Galileo; and step forward Bible in hand (like the profane Israelites carrying the Ark of God into the field of battle) to check the inquiries of the Geologist, the Astronomer, or the Political-economist, from an apprehension that the cause of religion can be endangered by them. [See First Lecture on Political Economy.] Any theory on whatever subject, that is really sound, can never be inimical to a religion founded on truth; and any that is unsound may be refuted by arguments drawn from observation and experiment, without calling in the aid of revelation. If we give way to a dread of danger from the inculcation of any scriptural doctrine, or from the progress of physical or moral science, we manifest a want of faith in God's power, or in his will, to maintain his own cause. That we shall indeed best further his cause by fearless perseverance in an open and straight course, I am firmly persuaded; but it is not only when we perceive the mischiefs of falsehood and disguise, and the beneficial tendency of fairness and candor, that we are to be followers of truth: the trial of our faith is, when we cannot perceive this: and the part of a lover of truth is to follow her at all seeming hazards, after the example of Him who 'came into the world that He might bear witness to the Truth.'" [Sermons by the Archbishop of Dublin.]

Appendix

Appendix Note A, On the great law Which Regulates Matter

Ever since the period when Newton established the great law of gravity, philosophers have occasionally speculated on the existence of some more comprehensive law, of which gravity itself is a consequence. Although some have considered it vain to search for a more general law, the great philosopher himself left encouragement to future inquirers; and the time, perhaps, has even now arrived, when such a discovery may be near its maturity. It would occupy too much space to introduce many illustrations of this opinion; there is, however, one which deserves attention, because it is not merely a happy conjecture, but the hypothesis on which it rests has been carried out by its author, through the aid of profound mathematical reasoning, to many of its remote consequences.

M. Mosotti [Professor of Physics at the University of the Ionian Islands] has shown, that by supposing matter to consist of two sorts of particles, each of which repels similar particles, directly as the mass, and inversely as the squares of their distances, whilst each attracts those of the other kind, also according to the same law,— then the resulting attractions explain all the phenomena of electricity, while there remains a residual force, acting at all sensible distances, according to the law of gravity.

Many of the discoveries of the present day point towards some more general law; and many philosophers of the present time anticipate its near approach. Under these circumstances, it may be interesting as well as useful briefly to state the principles which such a law must comprehend; and to indicate, however imperfectly, the path to be pursued in the research.

If matter be supposed to consist of two sorts of particles, or rather, perhaps, of two sorts of centers of force, of different orders of density; and if the particles of each order repel their own particles, according to a given law, but attract particles of the other kind, according to another law, — then, if we conceive only one particle of the denser kind to exist, and an infinite number of the other kind, that single particle will become the centre of a system, surrounded by all the others, which will form around it an atmosphere denser near the central body.

If we conceive a stream of particles, similar to those forming the atmosphere, to impinge upon it, so as just to overcome its resistance, they will, whilst continually producing undulations throughout its whole extent, gradually increase its magnitude, until it attains such a size, that the repulsion of the particles at the outer surface of this enlarged atmosphere is just balanced by the attraction of the central particle. If the stream continue after this point is reached, the whole outer layer will be pressed a little beyond the limit of attraction, and will fly off at right angles to the surface, which might then be said to radiate.

If the whole of the space in which such a central particle with its atmosphere is placed, is itself full of atmospheric particles, then their density will increase in approaching the central body; and if a stream of such particles were directed towards the center, they might produce throughout the atmosphere vibrations, which would be transmitted from it in all directions.

If two such central particles, with their atmospheres, exist at a distance from each other, they will be drawn together by a force depending on the difference between the mutual repulsion of their atmospheres and central bodies respectively for each other, and the attraction of each central particle for its neighbor's atmosphere: and in order to coincide with the existing law of nature, this force must be directly as the mass, and inversely as the square, of the distance. The other conditions which such a law must satisfy, are —

1. That the juxtaposition of such atoms must, in some circumstances, form a solid body: —

2. In other circumstances, a fluid.

3. That again, in still other circumstances, its particles shall repel each other, or the body become gaseous.

4. In the first state the body must possess cohesion, tenacity, malleability, elasticity; the measure and extent of each of which must result generally from the original law, and in each particular case from the constants belonging to the substance itself.

5. In the second state, it must possess capillarity, susceptibility of being compressed without becoming solid, as also elasticity.

But besides these, the central atoms must admit of a more intimate approach, so that their atmospheres may unite and form one atmosphere. This might constitute chemical union. Binary compounds might then (supposing the distance between the two central particles to be very small, compared with the diameters of the atmospheres) have atmospheres not quite spherical, and attracting differently in different directions; thus possessing polarity. Combinations of three or more atoms, as the central body of one atmosphere, might give great varieties of attractive forces. Each different combination would give a different atmosphere; and the equation of its surface might, perhaps, become the mathematical expression of the substance it constituted. Thus, all the phenomena produced by bodies, acting chemically on each other, might be deduced from the comparison of the characteristic surfaces of the atmospheres of their atoms. Another result, also, might ensue. Two or more central atoms uniting, might either not be able to retain the same amount of atmosphere, or they might possibly be able to retain a larger quantity. If the particles of such atmospheres constituted heat, it would in the former case be given out, and in the latter absorbed by chemical union.

Hence the whole of chemistry, and with it crystallography, would become a branch of mathematical analysis, which, like astronomy, taking its constants from observation, would enable us to predict the character of any new compound, and possibly indicate the source from which its formation might be anticipated.

For the sake of simplicity, two species of particles only have been mentioned above; but it seems more probable, that matter consists of at least three kinds.

Suppose each of the three kinds to repel its own particles; and the central atom, whilst it repels similar particles, to attract those of the two other kinds; and moreover, that the latter are either repulsive, or indifferent to each other. We might then conceive matter to be made up of particles, each having a central point, with an atmosphere surrounding it, and this atmosphere again enclosed within another and larger one.

Under such circumstances, the outer atmosphere might give rise to heat and light, to solidity and fluidity, and the gaseous condition; to capillarity, to elasticity, tenacity, and malleability. The more intimate union of the central atoms, by which two or more become enclosed in one common atmosphere of the second kind, might represent chemical combinations, and perhaps that atmosphere itself be electricity. Possibly, also, this intermediate atmosphere, acted on by the pressure of the external one, and by the attraction of the central atom, might take the liquid form. These binary or multiple-combinations of the original atoms, and their smaller atmospheres, would still be enclosed in an atmosphere of the outer kind, which might be nearly spherical. The joint action of the three might, at sensible distances, produce gravity.

The reader should, however, bear in mind, that these hints are thrown out only as objects of reflection and inquiry; and that nothing but a profound mathematical investigation can establish them, or even give to them that temporary value which arises from any hypothesis, representing a large collection of facts.

Appendix Note B, On the Calculating Engine

The nature of the arguments advanced in this volume having obliged me to refer, more frequently than I should have chosen, to the Calculating Engine, it becomes necessary to give the reader some brief account of its progress and present state.

About the year 1821, I undertook to superintend, for the Government, the construction of an engine for calculating and printing mathematical and astronomical tables. Early in the year 1833, a small portion of the machine was put together, and was found to perform its work with all the precision which had been anticipated. At that period circumstances, which I could not control, caused what I then considered a temporary suspension of its progress; and the Government, on whose decision the continuance or discontinuance of the work depended, have not yet communicated to me their wishes on the question. The first illustration I have employed is derived from the calculations made by this engine.

About October, 1834, I commenced the design of another, and far more powerful engine. Many of the contrivances necessary for its performance have since been discussed and drawn according to various principles; and all of them have been invented in more than one form. I consider them, even in their present state, as susceptible of practical execution; but time, thought, and expense, will probably improve them. As the remaining illustrations are all drawn from the powers of this new engine, it may be right to state, that it will calculate the numerical value of any algebraical function — that, at any period previously fixed upon, or contingent on certain events, it will cease to tabulate that algebraic function, and commence the calculation of a different one, and that these changes may be repeated to any extent.

The former engine could employ about 120 figures in its calculations; the present machine is intended to compute with about 4,000.

Here I should willingly have left the subject; but the public having erroneously imagined, that the sums of money paid to the workmen for the construction of the engine, were the remuneration of my own services, for inventing and directing its progress; and a Committee of the House of Commons having incidentally led the public to believe that a sum of money was voted to me for that purpose, — I think it right to give to that report the most direct and unequivocal contradiction.

Appendix Note C, Extract from the Theory of Probabilities of LaPlace

(For complete footnotes and appendices, including the writings of other authors in French, see the Victorian Web, where this treatise is presented in full.)

Appendix Note D, On Miracles

The view taken of miracles in Chapter VIII. is the same as that contained in the work of Butler, on the Analogy of Religion to the Constitution and Course of Nature. Inquiries connected with the Calculating Engine, impressed it very forcibly on my own mind, and I have drawn the illustrations chiefly from that subject. I cannot, however, forbear referring the reader to the opinion of Sir J. Herschel, expressed at the beginning of his letter to Mr. Lyell, (see Note I. p. 225,) because it confirms me in the belief, that the more profoundly we inquire into the mechanism of nature, the more certainly we arrive at that conclusion.

Appendix Note E, On Hume's Argument against Miracles

The reader will observe, that throughout the chapter to which this note refers, as well as in the note itself, the argument of Hume is taken strictly according to his own interpretation of the terms he uses, and the calculations are founded on them; so that it is from the very argument itself, when fairly pursued to its full extent, that the refutation results.

Both our belief in the truth of human testimony, and our belief in the permanence of the laws of nature, are, according to Hume, founded on experience; we may, therefore, in the complete ignorance in which he assumes we are, with respect to the causes of either, treat the question as one of the probability of an event deduced solely from observations of the past.

The argument of Hume asserts, that one improbability, namely, that of the falsehood of the testimony in favor of a miracle, must always be greater than another improbability, namely, that of the occurrence of the miracle itself; and also, that, from the very nature of human experience, this preponderance can never take place.

Now the only possible mode of disproving the assertion, that one thing cannot, under any circumstances, be greater than another, is to measure, under all circumstances, the numerical value of the two things so compared, and the truth or falsehood of the assertion will then appear. The doctrine of chances, which has been much improved since the time of Hume, now enables us to apply precise measures to this argument; and it is the object of this Note to state the outlines of the calculation, and the results to which it leads. Previously to this, however, it may not be amiss to offer a few remarks on the principles about to be employed.

In the great work of Laplace, "Théorie Analytique des Probabilités," those principles are established, and they are not merely undisputed, but are admitted by other writers of the highest authority on this subject. They form a part of the received knowledge of the present day, and, as such, they are employed in the present work, in which I propose to use, not to discuss them. I state this, because it has occasionally been asserted by persons unacquainted with the doctrine of chances, that the argument respecting the probability or improbability of miracles does not admit of the application of numbers. The received foundations of science are not to be put aside by such opinions, however highly skilled their authors may be in other branches of knowledge, and however powerful the intellect by which they may have attained those acquirements. The conclusions arrived at by the application of pure analysis must ever rest on the truth of the principles assumed at the commencement of the inquiry; and although a knowledge of mathematics may not appear necessary for forming a right judgment of the accuracy of those principles, yet it is observed, that a clear apprehension of them is not often found in the minds of those who are unacquainted with that science. When, however, the grounds on which the principles employed in the doctrine of chances are called in question by competent authority, it will be time enough to examine the question; and none will more eagerly enter upon that examination than those best versed in it, for none are so well aware of the extreme difficulty and delicacy of the subject.

As confusion sometimes arises from the difference in the meaning of the words probable and improbable in popular language and in mathematical inquiries, it may be convenient to point it out; and to state, that in this Note it is used in the mathematical sense, unless the reader's attention is directly called to a question relating to its popular sense.

In common language, an event is said to be probable when it is more likely to happen than to fail: it is said to be improbable when it is more likely to fail than to happen.

Now, an event whose probability is, in mathematical language 1/p, will be called probable or improbable, in ordinary language, according as p is less or greater than 2.

If, in mathematical language, 1/p expresses the probability of an event happening, 1—1/p expresses the probability of its failing, or the improbability of its happening.

It has been stated in the text, that two views may be taken of those extraordinary deviations from the usual course of nature, called miracles. According to the first of these, we have to calculate the probability that a white ball has been drawn from an urn (containing only white and black balls, out of which m balls have been drawn all black), as deduced from the testimony of witnesses whose probability of speaking truth is known:— or, of the analogous case; it having been observed that m persons have died without any restoration to life, what is the probability that such a resurrection has happened, it having been asserted by n independent witnesses, the probability of each of whose speaking false is 1/p?

The probability of the death without resurrection of the (m + 1)th is (m +1)/(m + 2), and the improbability of such an occurrence, independently of testimony, is 1/(m + 2); which is therefore the probability of a contrary occurrence, or that of a person being raised from the dead.

Now only two hypotheses can be formed, collusion being, by hypothesis, out of the question: either the event did happen, and the witnesses agree in speaking the truth, the probability of their concurrence being (1 — 1/p)ⁿ, and of that of the hypothesis being 1/(m + 2); or the event did not happen, and the witnesses agree in a falsehood, the probability of their concurrence being (1/p)ⁿ, and that of the hypothesis (m +1)/(m + 2).

The probability of the witnesses speaking truth, and the event occurring, is therefore,

and the probability of their falsehood is,

If we interpret Hume's assertion, "that the falsehood of the witnesses must be more improbable than the occurrence of the miracle," according to the mathematical meaning of the word improbable, then we must have,

(m +1)/((p + 1) + m + 1) < 1/(m + 2);

or,

(m + 1).(m + 2) < (p — 1)ⁿ + m + 1);

hence,

(p — 1)ⁿ + > (m + 1).(m + 2) — (m + 1) > (m + 1)²,

from which we find,

n > (2 log.(m + 1)/log.(p — 1).

If p is any number greater than two, this equation can always be satisfied.

It follows, therefore, that, however large m may be, or however great the quantity of experience against the occurrence of a miracle, (provided only that there are persons whose statements are more frequently correct than incorrect, and who give their testimony in favor of it without collusion,) a certain number n can ALWAYS be found; so that it shall be a greater improbability that their unanimous statement shall be a falsehood than that the miracle shall have occurred.

Let us now suppose each witness to state one falsehood for every ten truths p = 11, and m = 1000,000,000,000;

then, n > 2 log. (10¹² + 1)/log.10 > 24.

or twenty-five such witnesses are sufficient.

If the witnesses only state one falsehood for every hundred truths, then thirteen such witnesses are sufficient.

Another view of the question might be taken; and it might be asserted that, in order to believe in the miracle, the probability of its truth must be greater than the probability of its falsehood; in this case the expression (A) must be greater than (B) or,

(p — 1)ⁿ/(p — 1)ⁿ + m + 1 > m + 1/(p — 1)ⁿ + m + 1;

hence,

(p — 1)ⁿ > m + 1,

and

n > log.(m + 1)/log.(p — 1).

In this case also, under the same circumstances, the condition can always be fulfilled by by finding a sufficient number of witnesses to render the miracle probable, or even to give it any required degree of probability.

If p = 11, and m = 1000,000,000,000, as before, then,

n > log.(10¹²) + 1)/log.10 > 12.

According to the second view stated in the text, a miracle may be assimilated to the drawing of a given number i out of an urn, containing all numbers from 1 to m.

In this case the probability of the occurrence of the event is 1/m, and the probability of the concurrence of n witnesses in falsehood is (1/p)ⁿ.

Hence the probability that the particular number i was drawn, as deduced from the testimony of n witnesses, each of whose probability of falsehood is 1/p, is expressed by,

and the probability of the number i not having been drawn, or of their falsehood, is

Hence the improbability of the testimony must, according to Hume, be greater than that of the occurrence of the event; or,

If it is only required that the probability of the occurrence of the miracle shall be greater than its improbability, then we must make (C) greater than (D); or,

Hence in this view, also, a sufficient number of witnesses of given veracity may always be found to render the improbability of their concurrent independent testimony being false, greater than the improbability of the occurrence of the miracle.

There is, however, one other view, which it seems probable would have been that taken by Hume himself, had he applied numbers to his own argument. Considering the probability of the coincidence in falsehood of n persons each having the probability (p — 1)/p in favor of his truth, which is 1/pⁿ, that probability ought to be less than that of the occurrence of the miracle; or,

1/pⁿ < 1/(m + 2);
hence pⁿ > m + 2,
or, n > log.(m + 2)/log.p.

According to this view also, if m = 1000,000,000,000, and p = 11 > 12/1.04 > 12.

This view of the question refers to the probability of the concurrence of the witnesses before they have given their testimony. The other four cases relate to the probability of the miracle having happened, as deduced from the fact of the testimony having been given. The last seems to have been that which Hume would have himself arrived at; the others represent the true methods of estimating the probabilities of the various cases: and the important conclusion follows, that, whichever be the interpretation given to the argument of Hume, if independent witnesses can be found, who speak truth more frequently than falsehood, it is always possible to assign a number of independent witnesses, the improbability of the falsehood of whose concurring testimony shall be greater than that of the improbability of the miracle itself.

It is to be observed, that the whole of this argument applies to independent witnesses. The possibility of the collusion, and the degree of credit to be assigned to witnesses under any given circumstances, depend on facts which have not yet been sufficiently collected to become the subject of mathematical inquiry. Some of those considerations which bear on this part of the subject, the reader will find treated of in the work of Dr. Conyers Middleton, entitled "A Free Inquiry into the Miraculous Powers which are supposed to have subsisted in the Christian Church, from the earliest Ages through several successive Centuries." London, 1749.

Appendix Note F, On the Consequences of Central Heat

The increase of temperature observed as we descend below the earth's surface, as well as other phenomena, have led to a very general opinion, that great heat exists in the interior of the earth, and that the body of our planet, having been at one time intensely heated, has cooled down to its present temperature. With the view of pointing out courses of inquiry, by which these opinions may ultimately be tested by observation, it may be expedient to take a cursory view of some of the consequences of such an hypothesis.

And first, let us imagine the exterior of our globe to have once been in a state of intense heat. No fluid such as water could then have existed on the surface: it would instantly have been converted into vapor; and notwithstanding the increased weight of atmosphere thus produced and pressing on the surface of the globe, sufficient heat would reduce all fluids to the gaseous state. Let us, however, inquire as to the possible extent of such an atmosphere. In the first place, it could not extend beyond that point at which the moon's attraction is equal to that of the earth.

In the next place, much more contracted limits would be prescribed by the effect of centrifugal force, and of the cooling of the vapor by expansion, and by its distance from the source of radiant heat, which had caused its evaporation.

It would be interesting to inquire, what would be the nature of the surface of the atmosphere under such circumstances. At the distance at which the centrifugal force is equal to that of gravity, it might happen that the temperature was scarcely sufficient to maintain the water in a gaseous state. Should this have been the case, a belt of perpetual clouds might have been formed, resembling those of Jupiter. If, at this limit, a still lower degree of temperature prevailed, instead of a belt of clouds, a ring of ice might be formed.

This ring of ice, being exposed to different effects of radiation from variations in the radiating power of various parts of the earth's surface, might, by the superior heat at some parts, become diminished, whilst the condensation of the vapor might augment parts less exposed, and situated nearer to the body of the planet: and these conditions might continue, until at last the ring itself was melted or partially melted through at one or more points, and the whole might break up, and the fragments moving in a resisting medium, would ultimately fall down on the surface of the planet. The tearing up of that surface from such an event, would be augmented by the sudden conversion of the solid ice into steam; and after a time, the fragments of the ring would be absorbed again into the atmosphere of the planet.

Let us now suppose, owing to the gradual cooling down of the whole globe, the limit of condensation of steam into water, to occur at a nearer point than that at which the centrifugal force equals that of gravity. As soon as the steam is condensed into water, it will descend towards the surface of the earth; but that surface being still very hot, will, by its radiation, again convert the descending shower into steam; and this will happen at different heights above the surface, according to the radiating power of the part below. We may, therefore, conceive a shell surrounding the earth, the outer surface of which has just been condensed into water, and the inner consists of vapor, just re-converted into that state by the earth's radiation. These surfaces will attain different heights in different places. Between these two surfaces there will exist a perpetual rain, descending from the upper as a gentle shower, becoming gradually a violent torrent, and then as it falls re-absorbed into another gentle shower, which is entirely converted into vapor in approaching the heated surface.

Such being the state of things, let us imagine the globe to cool down uniformly. The lower surface of the descending rain, which is placed at irregular heights, will at length be brought down to the earth's surface in one or more points. The effect of this, which will in the first instance be a gentle shower, would be to cool that portion of the surface on which it falls, and hence to diminish its radiating power. This change, in its turn, will lower the under surface of the watery shell, so that a more violent rain, and ultimately an impetuous torrent will continue, perhaps, for thousands of years, its unremitted vertical action on the surface exposed to its force. The excavation of the largest valleys, or even of ocean beds, is not too much to expect from such forces.

But let us take another view of the consequences of such an original state of incandescence. The whole of the fluids now on the surface of the earth must then have been suspended in its atmosphere. But the extent of that atmosphere is itself limited by various causes: the attraction of other bodies, the effects of centrifugal force, the decrease of temperature, and the distances at which the particles of gaseous bodies cease to repel each other, all have their influence in determining its form and magnitude. Let us suppose that we possessed data from which the approximate amount of vapor contained in the entire atmosphere were known, and consequently the whole quantity of water in it; then, since we know the area of the present seas, we might easily ascertain their average depth. If the result of such a computation should give a mean depth much less than that which we know the ocean to possess, — as, for instance, only a hundred feet, — then we might conclude, either that the surface of the earth had never been in such a state of incandescence as has been supposed, or if it had, that a new source of aqueous vapor had been supplied to it, subsequently to its cooling down.

Appendix Note G, On the action of existing causes in producing elevations and subsidences in portions of the earth's surface

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Appendix Note H. Two Tables showing expansion of granite when heated

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Appendix Note I. Extracts from two letters of Sir J. Herschel

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Appendix Note K. On the Elevation of Beaches by Tides

If the earth were a spheroid of revolution, covered by one uniform ocean, two great tidal waves would follow each other round the globe at a distance of twelve hours.

Suppose several high narrow strips of land were now to encircle the globe, passing through the opposite poles, and dividing the earth's surface into several great unequal oceans, a separate tide would be raised in each. When the tidal wave had reached the farthest shore of one of them, conceive the causes that produce it to cease; then the wave thus raised would recede to the opposite shore, and continue to oscillate until destroyed by the friction of its bed. But if, instead of ceasing to act, the causes which produced the tide were to reappear at the opposite shore of the ocean, at the very moment when the reflected tide had returned to the place of its origin; then the second tide would act in augmentation of the first, and, if this continued, tides of great height might be produced for ages. The result might be, that the narrow ridge dividing the adjacent oceans would be broken through, and the tidal wave traverse a broader tract than in the former ocean. Let us imagine the new ocean to be just so much broader than the old, that the reflected tide would return to the origin of the tidal movement half a tide later than before: then, instead of two superimposed tides, we should have a tide arising from the subtraction of one from the other. The alterations of the height of the tides on shores so circumstanced, might be very small; and this might again continue for ages: thus causing beaches to be raised at very different elevations, without any real alteration in the level either of the sea or land.

If we consider the superposition of derivative tides, similar effects might be found to result; and it deserves inquiry, whether it may not be possible to account for some remarkable and well-attested phenomena by such means.

The gradual elevation during the past century, of one portion of the Swedish coast above the Baltic, is a recognized fact, and has lately been verified by Mr. Lyell. [See Phil. Trans. 1836.] It is not probable, from the form and position of that sea, that two tides should reach it distant by exactly half the interval of a tide, and thus produce a very small tide; nor is it likely that by the gradual but slow erosion of the longer channel, one tide should almost imperceptibly advance upon the other: but it becomes an interesting question to examine whether, in other places, under such peculiar circumstances, it might not be possible that a series of observations of the heights of tides at two distant periods, might give a different position for the mean level of the sea at places so situated.

If we conceive two tides to meet at any point, one of which is twelve hours later than the other, the elevation of the waters will arise from the joint influence of both. Let us suppose, that from the abrasion of the channel, the later tide arrives each time one-hundredth of a second earlier than before. After about 3,150 years, the high water of the earlier tide will coincide in point of time with the low water of the later tide: and the difference of height between high and low water will be equal to the difference of the height of the two tides, instead of to their sum, as it was at the first epoch.

If, in such circumstances, the two tides were nearly equal in magnitude, it might happen that on a coast so circumstanced, there would at one time be scarcely any perceptible tide; and yet, 3000 years after, the tide might rise 30 or 40 feet, or even higher; and this would happen without any change of relative height in the land and water during the intervening time. Possibly this view of the effects which may arise, either from the wearing down of channels, or the filling up of seas through which tides pass, may be applied to explain some of the phenomena of raised beaches, which are of frequent occurrence. Natural philosophers are at present not quite agreed upon the mode of determining the mean level of the ocean. Whether it is to be deduced from the averages between the highest and lowest spring tide, or from the averages of all the intermediate ones, or from the means of the instantaneous heights of the tide at all intervening periods — or by whatever other process, its true level is yet to be ascertained. It may, perhaps, also be useful to suggest that, besides the actual level of the sea at any particular place, it would be also desirable to ascertain whether the time of high water at given epochs is not itself a changeable quantity.

These reflections, however, are only thrown out with the view of exciting discussion on a subject involved at present in great mathematical difficulties, and possessing, at the same time, the highest practical importance.

Appendix Note L. On Ripple Mark

The small waves raised on the surface of the water, by the passage of a slight breeze, are called Ripple; and a series of marks, very similar in appearance, which are sometimes seen at low water on the flat part of a sea-beach formed of fine sand, are called ripple-marks. Such marks occur in various strata of stone, and at various depths below the solid surface of the globe, and are regarded as evidence of their having been formed beneath the sea. Similar appearances occur when a strong wind drives over the face of a sandy plain, and are frequently seen upon the surface of snow.

It appears that two fluids of different specific gravity, the lighter passing over the surface of the former, always concur in the formation of ripple. It seems also that the lines of ripple-mark are at right angles to the direction of the current which forms them.

If a fluid like air pass over the surface of perfectly quiescent water, in a plain absolutely parallel, it will have no effect; but if it impinge on the surface of the water with the slightest inclination, it will raise a small wave, which will be propagated by undulations to great distances. If the direction of the wind is very nearly parallel to the surface, this first wave, being raised above the general surface, will protect that part of the water immediately beyond it from the full effect of the wind, which will therefore again impinge upon the water at a little distance: and, this impact concurring with the undulation, will tend to produce another small wave, and thus, new waves will be produced. But the under surface of the air itself will also during this process assume the form of waves, and so, on the slightest deviation at any one point from absolute parallelism in the two fluids, their whole surfaces will become covered with ripples.

If one of the fluids be water, and the lower fluid be fine sand, partially suspended in water, these marks do not disappear when the cause ceases to act, as they do when formed by movement of air over the surface of water; but they remain and form the ridges or ripple which we observe when the tide has receded from a flat, sandy shore.

If, after the formation of ripple-marks at the bottom of a shallow sea, some adjacent river or some current deposit upon them the mud which it holds in suspension, then the first marks will be preserved, and new ripple marks may appear above them. Such is the origin of those marks we observe in various sand-stones, from the most recent down to those of the coal measures.

Dr. Fitton informs me, that the sand hills on the south of Etapes (in France) consist of ripple-marks produced by the wind on a very large scale. They are crescent-shaped hillocks, many of which are more than a hundred feet high. The height is greatest in the middle of the crescents, declining towards the points; and the slope on the inner side of the crescent, which is remote from the prevailing direction of the winds, is much more rapid than that on which it strikes.

Mr. Lyell has observed and described this mode of formation of ripple on the dunes of sand near Calais; remarking, that in that case there is an actual lateral transfer; the grains of sand being carried by the wind up the less inclined slope of the ripple, and falling over the steep scarp. I have observed the same fact at Swansea.

A similar explanation seems to present itself as the origin of that form of clouds familiarly known as "a mackerel sky" — a wave-like appearance, which probably arises from the passage of a current of air above or below a thin stratum of clouds. The air, being of nearly the same specific gravity as that of the cloud it acts upon, would produce ripple of larger size than would otherwise occur.

The surface of the sun presents to very good telescopes a certain mottled appearance, which is not exactly ripple, and which it is difficult to convey by description. It may, however, be suggested, that wherever such appearances occur, whether in planetary or in stellar bodies, or in the minuter precincts of the dye-house and the engine-boiler, they indicate the fitness of an inquiry, whether there are not two currents of fluid or semi-fluid matter, one moving with a different velocity over the other, the direction of the motion being at right angles to the lines of waves.

Appendix Note M. On the Age of Strata, as Inferred from the Rings of Trees Embedded in Them

The indelible records of past events which are preserved within the solid substance of our globe, may be in some measure understood without the aid of that refined analysis on which a complete acquaintance with them depends. The remains of vegetation, and of animal life, embedded in their coeval rocks, attest the existence of far distant times; and as science and the arts advance, we shall be enabled to read the minuter details of their living history. The object of the present note is to suggest to the reader a line of inquiry, by which we may still trace some small portion of the history of the past in the fossil woods which occur in so many of our strata.

It is well known that dicotyledonous trees increase in size by the deposition of an additional layer annually between the wood and the bark, and that a transverse section of such trees presents a series of nearly concentric though irregular rings, the number of which indicates the age of the tree. The relative thickness of these rings depends on the more or less flourishing state of the plant during the years in which they were formed. Each ring may, in some trees, be observed to be subdivided into others, thus indicating successive periods of the same year during which its vegetation was advanced or checked. These rings are disturbed in certain parts by irregularities resulting from branches; and the year in which each branch first sprung from the parent stock may be ascertained by proper sections.

It has been found by experiment, that even the motion imparted to a tree by the winds has an influence on its growth. Two young trees of equal size and vigor were selected and planted in similar circumstances, except that one was restrained from having any motion in the direction of the meridian, by two strong ropes fixed to it, and connecting it to the ground, at some distance towards the north and south. The other tree was by similar means prevented from having any motion in the direction of east and west. After several years, both trees were cut down, and the sections of their stems were found to be oval; but the longer axis of the oval of each was in the direction in which it had been capable of being moved by the winds.

These prominent effects are obvious to our senses, but every shower that falls, every change of temperature that occurs, and every wind that blows, leaves on the vegetable world the traces of its passage; slight, indeed, and imperceptible, perhaps, to us, but not the less permanently recorded in the depths of those woody fabrics. All these indications of the growth of the living tree are preserved in the fossil trunk, and with them also frequently the history of its partial decay.

Let us now inquire into the use we may make of these details relative to individual trees, when examining forests submerged by seas, embedded in peat mosses, or transformed, as in some of the older strata, into stone. Let us imagine, that we possessed sections of the trunks of a considerable number of trees, such as those occurring in the bed called the Dirt-bed [the reader will find an account of these fossil trees, and of the strata in which they occur, in several papers by Mr. Webster, Dr. Buckland, Mr. De la Beche, and Dr. Fitton, in the Transactions of the Geological Society of London, vol. iv. Series 2] in the island of Portland. If we were to select a number of trees of about the same size, we should probably find many of them to have been contemporaries. This fact would be rendered probable if we observed, as we doubtless should do, on examining the annual rings, that some of them conspicuous for their size occurred at the same distances of years in several trees. If, for example, we found on several trees a remarkably large annual ring, followed at the distance of seven years by a remarkably thin ring, and this again, after two years, followed by another large ring, we should reasonably infer that seven years after a season highly favorable to the growth of these trees, there had occurred a season peculiarly unfavorable to them; that after two more years another very favorable season had happened, and that all the trees so observed had existed at the same period of time. The nature of the season, whether hot or cold, wet or dry, might be conjectured with some degree of probability, from the class of tree under consideration. This kind of evidence, though slight at first, receives additional and great confirmation by the discovery of every new ling which supports it; and, by an extensive concurrence of such observations, the succession of seasons might be in some measure ascertained at remote geological periods.

On examining the shape of the sections of such trees, we might perceive some general tendency towards a uniform inequality in their diameters; and we should perhaps find that the longer axes of the sections most frequently pointed in one direction. If we knew from the species of tree that it possessed no natural tendency to such an inequality, then we might infer that, during the growth of these trees, they were bent most frequently in one direction; and hence derive an indication of the prevailing winds at that time. In order to determine from which of the two opposite quarters these winds came, we might observe the centers of these sections; and we should generally find that the rings on one side were closer and more compressed than those on the opposite side. From this we might infer the most exposed side, that from which the wind most frequently blew. Doubtless there would be many exceptions arising from local circumstances — some trees might have been sheltered from the direct course of the wind, and have only been acted upon by an eddy. Some might have been protected by adjacent large trees, sufficiently near to shelter them from the ruder gales, but not close enough to obstruct the light and air by which they were nourished. Such a tree might have a series of large and rather uniform rings, during the period of its protection by its neighbor; and these might be followed by a series of stinted and irregular ones, occasioned by the destruction of its protector. The same storm might have mutilated some trees, and half uprooted others: these latter might strive to support themselves for years, making but little addition, by stinted layers, to the thickness of their stems; and then, having thrown out new roots, they might regain their former rate of growth, until a new tempest again shook them from their places. Similar effects might result from floods and the action of rivers on the trees adjacent to their banks. But the effect of all these local and peculiar circumstances would disappear, if a sufficient number of sections could be procured from fossil trees, spread over a considerable extent of country.

The annual rings might however furnish other intimations of the successive existence of the trees.

On examining some rings remarkable for their size and position, let us suppose that we find, in one section, two remarkably large rings, separated from another large ring, by one very stinted ring, and this followed, after three ordinary rings, by two very small and two very large ones. Such a group might be indicated by the letters —

oLLsooosLLoo

where o denotes an ordinary year or ring, L a large one, and s a small or stinted ring. If such a group occurred in the sections of several different trees, it might fairly be attributed to general causes.

Let us now suppose such a group to be found near the center of one tree, and towards the external edge or bark of another; we should certainly conclude, that the tree near whose bark it occurred was the more ancient tree; that it had been advanced in age when that group of seasons occurred which had left their mark near the pith of the more recent tree, which was young at the time those seasons happened. If, on counting the rings of this younger tree, we found that there were, counting inward from the bark to this remarkable group, three hundred and fifty rings, we should justly conclude that, three hundred and fifty years before the death of this tree, which we will call A, the other, which we will call B, and whose section we possess, had then been an old tree. If we now search towards the center of the second tree B, for another remarkable group of rings; and if we also find a similar group near the bark of a third tree, which we will call C; and if, on counting the distance of the second group from the first in B, we find an interval of 430 rings, then we draw the inference that the tree A, 350 years before its destruction, was influenced in its growth by a succession of ten remarkable seasons, which also had their effect on a neighboring tree B, which was at that time of a considerable age. We conclude farther, that the tree B was influenced in its youth, or 420 years before the group of the ten seasons, by another remarkable succession of seasons, which also acted on a third tree, C, then old. Thus we connect the time of the death of the tree A with the series of seasons which affected the tree in its old age, at a period 770 years antecedent. If we could discover other trees having other cycles of seasons, capable of identification, we might trace back the history of the ancient forest, and possibly find in it some indications for conjecturing the time occupied in forming the stratum in which it is embedded.

The application of these principles to ascertaining the age of submerged forests, or to that of peat mosses, may possibly connect them ultimately with the chronology of man. Already we have an instance of a wooden hut with a stone hearth before it, and burnt wood on it, and a gate leading to a pile of wood, discovered at a depth of fifteen feet below the surface of a bog in Ireland: and it was found that this hut had probably been built when the bog had only reached half its present thickness, since there were still fifteen feet of turf below it.

The realization of the views here thrown out would require the united exertions of many individuals patiently exerted through a series of years. The first step must be to study fully the relations of the annual rings in every part of an individual tree. The effect of a favorable or unfavorable season on a section near the root must be compared with the influence of the same circumstance on its growth towards the top of the tree. Vertical sections also must be examined in order to register the annual additions to its height, and to compare them with its increase of thickness. Every branch must be traced to its origin, and its sections be registered. The means of identifying the influence of different seasons in various sections of the same individual tree and its branches being thus attained, the conclusions arrived at must be applied to several trees under similar circumstances, and such modifications must be applied to them as the case may require; and before any general conclusions can be reached respecting a tract of country once occupied by a forest, it will be necessary to have a considerable number of sections of trees scattered over various parts of it.

Appendix Note N. On a method of multiplying illustrations from woodcuts

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