file which includes the original illustrations4 ,see #bdbcf-;h4htm or #bdbcf-;h4zip3 7https3_/_/www4gutenberg4org_/dirs_/#b_/ d_/b_/c_/bdbcf_/bdbcf-;h_/#bdbcf-;h4htm7 or 7https3_/_/www4gutenberg4org_/dirs_/#b_/ d_/b_/c_/bdbcf_/bdbcf-;h4zip7 ,the ,romance of ,science4 ,,time ,,and ,,tide1 ,a ,romance of the ,moon4 ,being ,two ,lectures ,delivered in the ,theatre of the ,london ,institution1 on the ,afternoons of ,november #ai and #bf1 #ahhh4 by ,,sir ,,robert ,s4 ,,ball1 ,,ll4,d41 ,f4,r4,s41 ,royal ,astronomer of ,ireland4 ,published under the ,direction of the ,committee of ,general ,literature and ,education ,appointed by the ,society for ,promoting ,christian ,knowledge4 ,london3 ,society for ,promoting ,christian ,knowledge1 ,northumberland ,avenue1 ,w4,c42 #dc1 ,queen ,victoria ,street1 ,e4,c4 ,brighton3 #ace1 ,north ,street4 ,new ,york3 ,e4 ,j4 ,b4 ,young ,co4 ,illustration3 ,,view ,,of ,,the ,,moon ,,two ,,days ,,after ,,first ,,quarter4 ,from a photograph by ,mr4 ,,lewis ,m4 ,,rutherford4 ,frontispiece4 ,,to ,the ,members of the ,london ,institution ,i ,,dedicate ,,this ,,little ,,book4 ,,preface4 ,having been honoured once again with a request that ,i should lecture before the ,london ,institution1 ,i chose for my subject the ,theory of ,tidal ,evolution4 ,the kind reception which these lectures received has led to their publication in the present volume4 ,i have taken the opportunity to supplement the lectures as actually delivered by the insertion of some additional matter4 ,i am indebted to my friends ,mr4 ,close and ,mr4 ,rambaut for their kindness in reading the proofs4 ,,robert ,s4 ,,ball4 ,observatory1 ,,co4 ,,dublin1 ,april #bf1 #ahhi4 ,,time ,,and ,,tide4 ,,lecture ,i4 ,it is my privilege to address you this afternoon on a subject in which science and poetry are blended in a happy conjunction4 ,if there be a peculiar fascination about the earlier chapters of any branch of history1 how great must be the interest which attaches to that most primeval of all terrestrial histories which relates to the actual beginnings of this globe on which we stand4 ,in our efforts to grope into the dim recesses of this awful past1 we want the aid of some steadfast light which shall illumine the dark places without the treachery of the will-o'-the-wisp4 ,in the absence of that steadfast light1 vague conjectures as to the beginning of things could never be entitled to any more respect than was due to mere matters of speculation4 ,of late1 however1 the required light has been to some considerable extent forthcoming1 and the attempt has been made1 with no little success1 to elucidate a most interesting and wonderful chapter of an exceedingly remote history4 ,to chronicle this history is the object of the present lectures before this ,institution4 ,first1 let us be fully aware of the extraordinary remoteness of that period of which our history treats4 ,to attempt to define that period chronologically would be utterly futile4 ,when we have stated that it is more ancient than almost any other period which we can discuss1 we have expressed all that we are really entitled to say4 ,yet this conveys not a little4 ,it directs us to look back through all the ages of modern human history1 through the great days of ancient ,greece and ,rome1 back through the times when ,egypt and ,assyria were names of renown1 through the days when ,nineveh and ,babylon were mighty and populous cities in the zenith of their glory4 ,back earlier still to those more ancient nations of which we know hardly anything1 and still earlier to the prehistoric man1 of whom we know less2 back1 finally1 to the days when man first trod on this planet1 untold ages ago4 ,here is indeed a portentous retrospect from most points of view1 but it is only the commencement of that which our subject suggests4 ,for man is but the final product of the long anterior ages during which the development of life seems to have undergone an exceedingly gradual elevation4 ,our retrospect now takes its way along the vistas opened up by the geologists4 ,we look through the protracted tertiary ages1 when mighty animals1 now generally extinct1 roamed over the continents4 ,back still earlier through those wondrous secondary periods1 where swamps or oceans often covered what is now dry land1 and where mighty reptiles of uncouth forms stalked and crawled and swam through the old world and the new4 ,back still earlier through those vitally significant ages when the sunbeams were being garnered and laid aside for man's use in the great forests1 which were afterwards preserved by being transformed into seams of coal4 ,back still earlier through endless thousands of years1 when lustrous fishes abounded in the oceans2 back again to those periods characterized by the lower types of life2 and still earlier to that incredibly remote epoch when life itself began to dawn on our awakening globe4 ,even here the epoch of our present history can hardly be said to have been reached4 ,we have to look through a long succession of ages still antecedent4 ,the geologist1 who has hitherto guided our view1 cannot render us much further assistance2 but the physicist is at hand--he teaches us that the warm globe on which life is beginning has passed in its previous stages through every phase of warmth1 of fervour1 of glowing heat1 of incandescence1 and of actual fusion2 and thus at last our retrospect reaches to that particular period of our earth's past history which is specially illustrated by the modern doctrine of ,time and ,tide4 ,the present is the clue to the past4 ,it is the steady application of this principle which has led to such epoch-making labours as those by which ,lyell disclosed the origin of the earth's crust1 ,darwin the origin of species1 ,max ,mller the origin of language4 ,in our present subject the course is equally clear4 ,study exactly what is going on at present1 and then have the courage to apply consistently and rigorously what we have learned from the present to the interpretation of the past4 ,thus we begin with the ripple of the tide on the sea-beach which we see to-day4 ,the ebb and the flow of the tide are the present manifestations of an agent which has been constantly at work4 ,let that present teach us what tides must have done in the indefinite past4 ,it has been known from the very earliest times that the moon and the tides were connected together--connected1 ,i say1 for a great advance had to be made in human knowledge before it would have been possible to understand the true relation between the tides and the moon4 ,indeed1 that relation is so far from being of an obvious character1 that ,i think ,i have read of a race who felt some doubt as to whether the moon was the cause of the tides1 or the tides the cause of the moon4 ,i should1 however1 say that the moon is not the sole agent engaged in producing this periodic movement of our waters4 ,the sun also arouses a tide1 but the solar tide is so small in comparison with that produced by the moon1 that for our present purpose we may leave it out of consideration4 ,we must1 however1 refer to the solar tide at a later period of our discourses1 for it will be found to have played a very splendid part at the initial stage of the ,earth-,moon ,history1 while in the remote future it will again rise into prominence4 ,it will be well to set forth a few preliminary figures which shall explain how it comes to pass that the efficiency of the sun as a tide-producing agent is so greatly inferior to that of the moon4 ,indeed1 considering that the sun has a mass so stupendous1 that it controls the entire planetary system1 how is it that a body so insignificant as the moon can raise a bigger tide on the ocean than can the sun1 of which the mass is #bf1jjj1jjj times as great as that of our satellite8 ,this apparent paradox will disappear when we enunciate the law according to which the efficiency of a tide-producing agent is to be estimated4 ,this law is somewhat different from the familiar form in which the law of gravitation is expressed4 ,the gravitation between two distant masses is to be measured by multiplying these masses together1 and dividing the product by the square of the distance4 ,the law for expressing the efficiency of a tide-producing agent varies not according to the inverse square1 but according to the inverse cube of the distance4 ,this difference in the expression of the law will suffice to account for the superiority of the moon as a tide-producer over the sun4 ,the moon's distance on an average is about one #chfth part of that of the sun1 and thus it is easy to show that so far as the mere attraction of gravitation is concerned1 the efficiency of the sun's force on the earth is about one hundred and seventy-five times as great as the force with which the moon attracts the earth4 ,that is of course calculated under the law of the inverse square4 ,to determine the tidal efficiency we have to divide this by three hundred and eighty-six1 and thus we see that the tidal efficiency of the sun is less than half that of the moon4 ,when the solar tide and the lunar tide are acting in unison1 they conspire to produce very high tides and very low tides1 or1 as we call them1 spring tides4 ,on the other hand1 when the sun is so placed as to give us a low tide while the moon is producing a high tide1 the net result that we actually experience is merely the excess of the lunar tide over the solar tide2 these are what we call neap tides4 ,in fact1 by very careful and long-continued observations of the rise and fall of the tides at a particular port1 it becomes possible to determine with accuracy the relative ranges of spring tides and neap tides2 and as the spring tides are produced by moon plus sun1 while the neap tides are produced by moon minus sun1 we obtain a means of actually weighing the relative masses of the sun and moon4 ,this is one of the remarkable facts which can be deduced from a prolonged study of the tides4 ,the demonstration of the law of the tide-producing force is of a mathematical character1 and ,i do not intend in these lectures to enter into mathematical calculations4 ,there is1 however1 a simple line of reasoning which1 though it falls far short of actual demonstration1 may yet suffice to give a plausible reason for the law4 ,the tides really owe their origin to the fact that the tide-producing agent operates more powerfully on those parts of the tide-exhibiting body which are near to it1 than on the more distant portions of the same4 ,the nearer the two bodies are together1 the larger proportionally will be the differences in the distances of its various parts from the tide-producing body2 and on this account the leverage1 so to speak1 of the action by which the tides are produced is increased4 ,for instance1 if the two bodies were brought within half their original distance of each other1 the relative size of each body1 as viewed from the other1 will be doubled2 and what we have called the leverage of the tide-producing ability will be increased twofold4 ,the gravitation also between the two bodies is increased fourfold when the distance is halved1 and consequently1 the tide-producing ability is doubled for one reason1 and increased fourfold again by another2 hence1 the tides will be increased eightfold when the distance is reduced to one half4 ,now1 as eight is the cube of two1 this illustration may be taken as a verification of the law1 that the efficiency of a body as a tide-producer varies inversely as the cube of the distance between it and the body on which the tides are being raised4 ,for simplicity we may make the assumption that the whole of the earth is buried beneath the ocean1 and that the moon is placed in the plane of the equator4 ,we may also entirely neglect for the present the tides produced by the sun1 and we shall also make the further assumption that friction is absent4 ,what friction is capable of doing we shall1 however1 refer to later on4 ,the moon will act on the ocean and deform it1 so that there will be high tide along one meridian1 and high tide also on the opposite meridian4 ,this is indeed one of the paradoxes by which students are frequently puzzled when they begin to learn about the tides4 ,that the moon should pull the water up in a heap on one side seems plausible enough4 ,high tide will of course be there2 and the student might naturally think that the water being drawn in this way into a heap on one side1 there will of course be low tide on the opposite side of the earth4 ,a natural assumption1 perhaps1 but nevertheless a very wrong one4 ,there are at every moment two opposite parts of the earth in a condition of high water2 in fact1 this will be obvious if we remember that every day1 or1 to speak a little more accurately1 in every twenty-four hours and fifty one minutes1 we have on the average two high tides at each locality4 ,of course this could not be if the moon raised only one heap of high water1 because1 as the moon only appears to revolve around the earth once a day1 or1 more accurately1 once in that same average period of twenty-four hours and fifty-one minutes1 it would be impossible for us to have high tides succeeding each other as they do in periods a little longer than twelve hours1 if only one heap were carried round the earth4 ,the first question then is1 as to how these two opposite heaps of water are placed in respect to the position of the moon4 ,the most obvious explanation would seem to be1 that the moon should pull the waters up into a heap directly underneath it1 and that therefore there should be high water underneath the moon4 ,as to the other side1 the presence of a high tide there was1 on this theory1 to be accounted for by the fact that the moon pulled the earth away from the waters on the more remote side1 just as it pulled the waters away from the more remote earth on the side underneath the moon4 ,it is1 however1 certainly not the case that the high tide is situated in the simple position that this law would indicate1 and which we have represented in ,fig4 #a1 where the circular body is the earth1 the ocean surrounding which is distorted by the action of the tides4 ,illustration3 ,,fig4 #a4 ,we have here taken an oval to represent the shape into which the water is supposed to be forced or drawn by the tidal action of the tide-producing body4 ,this may possibly be a correct representation of what would occur on an ideal globe entirely covered with a frictionless ocean4 ,but as our earth is not covered entirely by water1 and as the ocean is very far from being frictionless1 the ideal tide is not the tide that we actually know2 nor is the ideal tide represented by this oval even an approximation to the actual tides to which our oceans are subject4 ,indeed1 the oval does not represent the facts at all1 and of this it is only necessary to adduce a single fact in demonstration4 ,i take the fundamental issue so often debated1 as to whether in the ocean vibrating with ideal tides the high water or the low water should be under the moon4 ,or to put the matter otherwise2 when we represent the displaced water by an oval1 is the long axis of the oval to be turned to the moon1 as generally supposed1 or is it to be directed at right angles therefrom8 ,if the ideal tides were in any degree representative of the actual tides1 so fundamental a question as this could be at once answered by an appeal to the facts of observation4 ,even if friction in some degree masked the phenomena1 surely one would think that the state of the actual tides should still enable us to answer this question4 ,but a study of the tides at different ports fails to realize this expectation4 ,at some ports1 no doubt1 the tide is high when the moon is on the meridian4 ,in that case1 of course1 the high water is under the moon1 as apparently ought to be the case invariably1 on a superficial view4 ,but1 on the other hand1 there are ports where there is often low water when the moon is crossing the meridian4 ,yet other ports might be cited in which every intermediate phase could be observed4 ,if the theory of the tides was to be the simple one so often described1 then at every port noon should be the hour of high water on the day of the new moon or of the full moon1 because then both tide-exciting bodies are on the meridian at the same time4 ,even if the friction retarded the great tidal wave uniformly1 the high tide on the days of full or change should always occur at fixed hours2 but1 unfortunately1 there is no such delightful theory of the tides as this would imply4 ,at ,greenock no doubt there is high water at or about noon on the day of full or change2 and if it could be similarly said that on the day of full or change there was high water everywhere at local noon1 then the equilibrium theory of the tides1 as it is called1 would be beautifully simple4 ,but this is not the case4 ,even around our own coasts the discrepancies are such as to utterly discredit the theory as offering any practical guide4 ,at ,aberdeen the high tide does not appear till an hour later than the doctrine would suggest4 ,it is two hours late at ,london1 three at ,tynemouth1 four at ,tralee1 five at ,sligo1 and six at ,hull4 ,this last port would be indeed the haven of refuge for those who believe that the low tide ought to be under the moon4 ,at ,hull this is no doubt the case2 and if at all other places the water behaved as it does at ,hull1 why then1 of course1 it would follow that the law of low water under the moon was generally true4 ,but then this would not tally with the condition of affairs at the other places ,i have named2 and to complete the cycle ,i shall add a few more4 ,at ,bristol the high water does not get up until seven hours after the moon has passed the meridian1 at ,arklow the delay is eight hours1 at ,yarmouth it is nine1 at the ,needles it is ten hours1 while lastly1 the moon has nearly got back to the meridian again ere it has succeeded in dragging up the tide on which ,liverpool's great commerce so largely depends4 ,nor does the result of studying the tides along other coasts beside our own decide more conclusively on the mooted point4 ,even ports in the vast ocean give a very uncertain response4 ,kerguelen ,island and ,santa ,cruz might seem to prove that the high tide occurs under the moon1 but unfortunately both ,fiji and ,ascension seem to present us with an equally satisfactory demonstration1 that beneath the moon is the invariable home of low water4 ,i do not mean to say that the study of the tides is in other respects such a confused subject as the facts ,i have stated would seem to indicate4 ,it becomes rather puzzling1 no doubt1 when we compare the tides at one port with the tides elsewhere4 ,the law and order are1 then1 by no means conspicuous1 they are often hardly discernible4 ,but when we confine our attention to the tides at a single port1 the problem becomes at once a very intelligible one4 ,indeed1 the investigation of the tides is an easy subject1 if we are contented with a reasonably approximate solution2 should1 however1 it be necessary to discuss fully the tides at any port1 the theory of the method necessary for doing so is available1 and a most interesting and beautiful theory it certainly is4 ,let us then speak for a few moments about the methods by which we can study the tides at a particular port4 ,the principle on which it is based is a very simple one4 ,it is the month of ,august1 the #ahth1 we shall suppose1 and we are going to enjoy a delicious swim in the sea4 ,we desire1 of course1 to secure a high tide for the purpose of doing so1 and we call an almanac to help us4 ,i refer to the ,thom's ,dublin ,directory1 where ,i find the tide to be high at #aj;h4 #adm4 on the morning of the #ahth of ,august4 ,that will then be the time to go down to the baths at ,howth or ,kingstown4 ,but what ,i am now going to discourse to you about is not the delights of sea-bathing1 it is rather a different inquiry4 ,i want to ask1 ,how did the people who prepared that almanac know years beforehand1 that on that particular day the tide would be high at that particular hour8 ,how do they predict for every day the hour of high water8 and how comes it to pass that these predictions are invariably correct8 ,we first refer to that wonderful book1 the ,nautical ,almanac4 ,in that volume the movements of the moon are set forth with full detail2 and among other particulars we can learn on page iv of every month the mean time of the moon's meridian passage4 ,it appears that on the day in question the moon crossed the meridian at #aa;h4 #bcm4 ,thus we see there was high water at ,dublin at #aj;h4 #adm41 and #a;h4 #im4 later1 that is1 at #aa;h4 #bcm41 the moon crossed the meridian4 ,let us take another instance4 ,there is a high tide at #c4dj ,p4,m4 on the #beth ,august1 and again the infallible ,nautical ,almanac tells us that the moon crossed the meridian at #e;h4 #ddm41 that is1 at #b;h4 #dm4 after the high water4 ,in the first case the moon followed the tide in about an hour1 and in the second case the moon followed in about two hours4 ,now if we are to be satisfied with a very rough tide rule for ,dublin1 we may say generally that there is always a high tide an hour and a half before the moon crosses the meridian4 ,this would not be a very accurate rule1 but ,i can assure you of this1 that if you go by it you will never fail of finding a good tide to enable you to enjoy your swim4 ,i do not say this rule would enable you to construct a respectable tide-table4 ,a ship-owner who has to creep up the river1 and to whom often the inches of water are material1 will require far more accurate tables than this simple rule could give4 ,but we enter into rather complicated matters when we attempt to give any really accurate methods of computation4 ,on these we shall say a few words presently4 ,what ,i first want to do1 is to impress upon you in a simple way the fact of the relation between the tide and the moon4 ,to give another illustration1 let us see how the tides at ,london ,bridge are related to the moon4 ,on ,jan4 #ast1 #ahhg1 it appeared that the tide was high at #f;h4 #bfm4 ,p4,m41 and that the moon had crossed the meridian #efm4 previously2 on the #hth ,jan4 the tide was high at #j;h4 #dcm4 ,p4,m41 and the moon had crossed the meridian #b;h4 #am4 previously4 ,therefore we would have at ,london ,bridge high water following the moon's transit in somewhere about an hour and a half4 ,i choose a day at random1 for example--the #abth ,april4 ,the moon crosses the upper meridian at #c;h4 #cim4 ,a4,m41 and the lower meridian at #d;h #fm4 ,p4,m4 ,adding an hour and a half to each would give the high tides at #e;h #im4 ,a4,m4 and #e;h4 #cfm4 ,p4,m42 as a matter of fact1 they are #d;h4 #ehm4 ,a4,m4 and #e;h4 #bjm4 ,p4,m4 ,but these illustrations are sufficient4 ,we find that at ,london1 in a general way1 high water appears at ,london ,bridge about an hour and a half after the moon has passed the meridian of ,london4 ,it so happens that the interval at ,dublin is about the same1 i4e4 an hour and a half2 only that in the latter case the high water precedes the moon by that interval instead of following it4 ,we may employ the same simple process at other places4 ,choose two days about a week distant2 find on each occasion the interval between the transit of the moon and the time of high water1 then the mean of these two differences will always give some notion of the interval between high water and the moon's transit4 ,if then we take from the ,nautical ,almanac the time of the moon's transit1 and apply to it the correction proper for the port1 we shall always have a sufficiently good tide-table to guide us in choosing a suitable time for taking our swim or our walk by the sea-side2 though if you be the captain of a vessel1 you will not be so imprudent as to enter port without taking counsel of the accurate tide-tables1 for which we are indebted to the ,admiralty4 ,every one who visits the sea-side1 or who lives at a sea-port1 should know this constant for the tides1 which affect him and his movements so materially4 ,if he will discover it from his own experience1 so much the better4 ,the first point to be ascertained is the time of high water4 ,do not take this from any local table2 you ought to observe it for yourself4 ,you will go to the pier head1 or1 better still1 to some place where the rise and fall of the mere waves of the sea will not embarrass you in your work4 ,you must note by your watch the time when the tide is highest4 ,an accurate way of doing this will be to have a scale on which you can measure the height at intervals of five minutes about the time of high water4 ,you will then be able to conclude the time at which the tide was actually at its highest point2 but even if no great accuracy be obtainable1 you can still get much interesting information1 for you will without much difficulty be right within ten minutes or a quarter of an hour4 ,the correction for the port is properly called the establishment1 this being the average time of high water on the days of full and change of the moon at the particular port in question4 ,we can considerably amend the elementary notion of the tides which the former method has given us1 if we adopt the plan described by ,dr4 ,whewell in the first four editions of the ,admiralty ,manual of ,scientific ,inquiry4 ,we speak of the interval between the transit of the moon and the time of high water as the luni-tidal interval4 ,of course at full and change this is the same thing as the establishment1 but for other phases of the moon the establishment must receive a correction before being used as the luni-tidal interval4 ,the correction is given by the following table-- ,hour of ,moon's transit after ,sun3 #j #a #b #c #d #e #f #g #h #i #aj #aa ---------------------------------------- ----- #j-bjm-#cjm-#ejm-#fjm-#fjm-#fjm-#djm-#aj m#ajm#bjm#ajm ,correction of establishment to find luni-tidal interval3 ,thus at a port where the establishment was #c;h4 #bem41 let us suppose that the transit of the moon took place at #f ,p4,m42 then we correct the establishment by -#fjm41 and find the luni-tidal interval to be #b;h4 #bem41 and accordingly the high water takes place at #h;h4 #bem4 ,p4,m4 ,but even this method is only an approximation4 ,the study of the tides is based on accurate observation of their rise and fall on different places round the earth4 ,to show how these observations are to be made1 and how they are to be discussed and reduced when they have been made1 ,i may refer to the last edition of the ,admiralty ,manual of ,scientific ,inquiry1 #ahhf4 ,for a complete study of the tides at any port a self-registering tide-gauge should be erected1 on which not alone the heights and times of high and low water should be depicted1 but also the continuous curve which shows at any time the height of the water4 ,in fact1 the whole subject of the practical observation and discussion and prediction of tides is full of valuable instruction1 and may be cited as one of the most complete examples of the modern scientific methods4 ,in the first place1 the tide-gauge itself is a delicate instrument2 it is actuated by a float which rises and falls with the water1 due provision being made that the mere influence of waves shall not make it to oscillate inconveniently4 ,the motion of the float when suitably reduced by mechanism serves to guide a pencil1 which1 acting on the paper round a revolving drum1 gives a faithful and unintermitting record of the height of the water4 ,thus what the tide-gauge does is to present to us a long curved line of which the summits correspond to the heights of high water1 while the depressions are the corresponding points of low water4 ,the long undulations of this curve are1 however1 very irregular4 ,at spring tides1 when the sun and the moon conspire1 the elevations rise much higher and the depressions sink much lower than they do at neap tides1 when the high water raised by the moon is reduced by the action of the sun4 ,there are also many minor irregularities which show the tides to be not nearly such simple phenomena as might be at first supposed4 ,but what we might hastily think of as irregularities are1 in truth1 the most interesting parts of the whole phenomena4 ,just as in the observations of the planets the study of the perturbations has led us to results of the widest interest and instruction1 so it is these minor phenomena of the tides which seem most pregnant with scientific interest4 ,the tide-gauge gives us an elaborate curve4 ,how are we to interpret that curve8 ,here indeed a most beautiful mathematical theorem comes to our aid4 ,just as ordinary sounds consist of a number of undulations blended together1 so the tidal wave consists of a number of distinct undulations superposed4 ,of these the ordinary lunar tide and the ordinary solar tide are the two principal2 but there are also minor undulations1 harmonics1 so to speak1 some originating from the moon1 some originating from the sun1 and some from both bodies acting in concert4 ,in the study of sound we can employ an acoustic apparatus for the purpose of decomposing any proposed note1 and finding not only the main undulation itself1 but the several superposed harmonics which give to the note its timbre4 ,so also we can analyze the undulation of the tide1 and show the component parts4 ,the decomposition is effected by the process known as harmonic analysis4 ,the principle of the method may be very simply described4 ,let us fix our attention on any particular tide1 for so the various elements are denoted4 ,we can always determine beforehand1 with as much accuracy as we may require1 what the period of that tide will be4 ,for instance1 the period of the lunar semi-diurnal tide will of course be half the average time occupied by the moon to travel round from the meridian of any place until it regains the same meridian2 the period of the lunar diurnal tide will be double as great2 and there are fortnightly tides1 and others of periods still greater4 ,the essential point to notice is1 that the periods of these tides are given by purely astronomical considerations from the periods of the motions theory1 and do not depend upon the actual observations4 ,we measure off on the curve the height of the tide at intervals of an hour4 ,the larger the number of such measures that are available the better2 but even if there be only three hundred and sixty or seven hundred and twenty consecutive hours1 then1 as shown by ,professor ,g4 ,h4 ,darwin in the ,admiralty ,manual already referred to1 it will still be possible to obtain a very competent knowledge of the tides in the particular port where the gauge has been placed4 ,the art 7for such indeed it may be described7 of harmonic analysis consists in deducing from the hourly observations the facts with regard to each of the constituent tides4 ,this art has been carried to such perfection1 that it has been reduced to a very simple series of arithmetical operations4 ,indeed it has now been found possible to call in the aid of ingenious mechanism1 by which the labours of computation are entirely superseded4 ,the pointer of the harmonic analyzer has merely to be traced over the curve which the tide-gauge has drawn1 and it is the function of the machine to decompose the composite undulation into its parts1 and to exhibit the several constituent tides whose confluence gives the total result4 ,as if nothing should be left to complete the perfection of a process which1 both from its theoretical and its practical sides1 is of such importance1 a machine for predicting tides has been designed1 constructed1 and is now in ordinary use4 ,when by the aid of the harmonic analysis the effectiveness of the several constituent tides affecting a port have become fully determined1 it is of course possible to predict the tides for that port4 ,each tide is a simple periodic rise and fall1 and we can compute for any future time the height of each were it acting alone4 ,these heights can all be added together1 and thus the height of the water is obtained4 ,in this way a tide-table is formed1 and such a table when complete will express not alone the hours and heights of high water on every day1 but the height of the water at any intervening hour4 ,the computations necessary for this purpose are no doubt simple1 so far as their principle is concerned2 but they are exceedingly tedious1 and any process must be welcomed which affords mitigation of a task so laborious4 ,the entire theory of the tides owes much to ,sir ,william ,thomson in the methods of observation and in the methods of reduction4 ,he has now completed the practical parts of the subject by inventing and constructing the famous tide-predicting engine4 ,the principle of this engine is comparatively simple4 ,there is a chain which at one end is fixed1 and at the other end carries the pencil which is pressed against the revolving drum on which the prediction is to be inscribed4 ,between its two ends the chain passes up and down over pulleys4 ,each pulley corresponds to one of the tides1 and there are about a dozen altogether1 some of which exercise but little effect4 ,of course if the centres of the pulleys were all fixed the pen could not move1 but the centre of each pulley describes a circle with a radius proportional to the amplitude of the corresponding tide1 and in a time proportional to the period of that tide4 ,when these pulleys are all set so as to start at the proper phases1 the motion is produced by turning round a handle which makes the drum rotate1 and sets all the pulleys in motion4 ,the tide curve is thus rapidly drawn out2 and so expeditious is the machine1 that the tides of a port for an entire year can be completely worked out in a couple of hours4 ,while the student or the philosopher who seeks to render any account of the tide on dynamical grounds is greatly embarrassed by the difficulties introduced by friction1 we1 for our present purpose in the study of the great romance of modern science opened up to us by the theory of the tides1 have to welcome friction as the agent which gives to the tides their significance from our point of view4 ,there is the greatest difference between the height of the rise and fall of the tide at different localities4 ,out in mid-ocean1 for instance1 an island like ,st4 ,helena is washed by a tide only about three feet in range2 an enclosed sea like the ,caspian is subject to no appreciable tides whatever1 while the ,mediterranean1 notwithstanding its connection with the ,atlantic1 is still only subject to very inconsiderable tides1 varying from one foot to a few feet4 ,the statement that water always finds its own level must be received1 like many another proposition in nature1 with a considerable degree of qualification4 ,long ere one tide could have found its way through the ,straits of ,gibraltar in sufficient volume to have appreciably affected the level of the great inland sea1 its effects would have been obliterated by succeeding tides4 ,on the other hand1 there are certain localities which expose a funnel-shape opening to the sea2 into these the great tidal wave rushes1 and as it passes onwards towards the narrow part1 the waters become piled up so as to produce tidal phenomena of abnormal proportions4 ,thus1 in our own islands1 we have in the ,bristol ,channel a wide mouth into which a great tide enters1 and as it hurries up the ,severn it produces the extraordinary phenomenon of the ,bore4 ,the ,bristol ,channel also concentrates the great wave which gives ,chepstow and ,cardiff a tidal range of thirty-seven or thirty-eight feet at springs1 and forces the sea up the river ,avon so as to give ,bristol a wonderful tide4 ,there is hardly any more interesting spot in our islands for the observation of tides than is found on ,clifton ,suspension ,bridge4 ,from that beautiful structure you look down on a poor and not very attractive stream1 which two hours later becomes transformed into a river of ample volume1 down which great ships are navigated4 ,but of all places in the world1 the most colossal tidal phenomena are those in the ,bay of ,fundy4 ,here the ,atlantic passes into a long channel whose sides gradually converge4 ,when the great pulse of the tide rushes up this channel1 it is gradually accumulated into a mighty volume at the upper end1 the ebb and flow of which at spring tides extends through the astonishing range of not less than fifty feet4 ,these discrepancies between the tides at different places are chiefly due to the local formations of the coasts and the sea-beds4 ,indeed1 it seems that if the whole earth were covered with an uniform and deep ocean of water1 the tides would be excessively feeble4 ,on no other supposition can we reasonably account for the fact that our barometric records fail to afford us any very distinct evidence as to the existence of tides in the atmosphere4 ,for you will1 of course1 remember that our atmosphere may be regarded as a deep and vast ocean of air1 which embraces the whole earth1 extending far above the loftiest summits of the mountains4 ,it is one of the profoundest of nature's laws that wherever friction takes place1 energy has to be consumed4 ,perhaps ,i ought rather to say transformed1 for of course it is now well known that consumption of energy in the sense of absolute loss is impossible4 ,thus1 when energy is expended in moving a body in opposition to the force of friction1 or in agitating a liquid1 the energy which disappears in its mechanical form reappears in the form of heat4 ,the agitation of water by paddles moving through it warms the water1 and the accession of heat thus acquired measures the energy which has been expended in making the paddles rotate4 ,the motion of a liquid of which the particles move among each other with friction1 can only be sustained by the incessant degradation of energy from the mechanical form into the lower form of diffused heat4 ,thus the very fact that the tides are ebbing and flowing1 and that there is consequently incessant friction going on among all the particles of water in the ocean1 shows us that there must be some great store of energy constantly available to supply the incessant draughts made upon it by the daily oscillation of the tides4 ,in addition to the mere friction between the particles of water1 there are also many other ways in which the tides proclaim to us that there is some great hoard of energy which is continually accessible to their wants4 ,stand on the bank of an estuary or river up and down which a great tidal current ebbs and flows2 you will see the water copiously charged with sediment which the tide is bearing along4 ,engineers are well aware of the potency of the tide as a vehicle for transporting stupendous quantities of sand or mud4 ,a sand-bank impedes the navigation of a river2 the removal of that sand-bank would be a task1 perhaps1 conceivably possible by the use of steam dredges and other appliances1 whereby vast quantities of sand could be raised and transported to another locality where they would be innocuous4 ,it is sometimes possible to effect the desired end by applying the power of the tide4 ,a sea-wall judiciously thrown out will sometimes concentrate the tide into a much narrower channel4 ,its daily oscillations will be accomplished with greater vehemence1 and as the tide rushes furiously backwards and forwards over the obstacle1 the incessant action will gradually remove it1 and the impediment to navigation may be cleared away4 ,here we actually see the tides performing a piece of definite and very laborious work1 to accomplish which by the more ordinary agents would be a stupendous task4 ,in some places the tides are actually harnessed so as to accomplish useful work4 ,i have read that underneath old ,london ,bridge there used formerly to be great water-wheels1 which were turned by the tide as it rushed up the river1 and turned again1 though in the opposite way1 by the ebbing tide4 ,these wheels were1 ,i believe1 employed to pump up water1 though it does not seem obvious for what purposes the water would have been suitable4 ,indeed in the ebb and flow all round our coasts there is a potential source of energy which has hitherto been allowed to run to waste4 ,the tide could be utilized in various ways4 ,many of you will remember the floating mills on the ,rhine4 ,they are vessels like paddle steamers anchored in the rapid current4 ,the flow of the river makes the paddles rotate1 and thus the machinery in the interior is worked4 ,such craft moored in a rapid tide-way could also be made to convey the power of the tides into the mechanism of the mill4 ,or there is still another method which has been employed1 and which will perhaps have a future before it in those approaching times when the coal-cellars of ,england shall be exhausted4 ,imagine on the sea-coast a large flat extent which is inundated twice every day by the tide4 ,let us build a stout wall round this area1 and provide it with a sluice-gate4 ,open the gate as the tide rises1 and the great pond will be filled2 then at the moment of high water close the sluice1 and the pond-full will be impounded4 ,if at low tide the sluice be opened the water will rush tumultuously out4 ,now suppose that a water-wheel be provided1 so that the rapid rush of water from the exit shall fall upon its blades2 then a source of power is obviously the result4 ,at present1 however1 such a contrivance would naturally find no advocates1 for of course the commercial aspect of the question is that which will decide whether the scheme is practicable and economical4 ,the issue indeed can be very simply stated4 ,suppose that a given quantity of power be required--let us say that of one hundred horse4 ,then we have to consider the conditions under which a contrivance of the kind we have sketched shall yield a power of this amount4 ,sir ,william ,thomson1 in a very interesting address to the ,british ,association at ,york in #ahha1 discussed this question1 and ,i shall here make use of the facts he brought forward on that occasion4 ,he showed that to obtain as much power as could be produced by a steam-engine of one hundred horse power1 a very large reservoir would be required4 ,it is doubtful indeed whether there would be many localities on the earth which would be suitable for the purpose4 ,suppose1 however1 an estuary could be found which had an area of forty acres2 then if a wall were thrown across the mouth so that the tide could be impounded1 the total amount of power that could be yielded by a water-wheel worked by the incessant influx and efflux of the tide would be equal to that yielded by the one hundred horse engine1 running continuously from one end of the year to the other4 ,there are many drawbacks to a tide-mill of this description4 ,in the first place1 its situation would naturally be far removed from other conveniences necessary for manufacturing purposes4 ,then too there is the great irregularity in the way in which the power is rendered available4 ,at certain periods during the twenty-four hours the mill would stop running1 and the hours when this happened would be constantly changing4 ,the inconvenience from the manufacturer's point of view of a deficiency of power during neap tides might not be compensated by the fact that he had an excessive supply of power at spring-tides4 ,before tide-mills could be suitable for manufacturing purposes1 some means must be found for storing away the energy when it is redundant1 and applying it when its presence is required4 ,we should want in fact for great sources of energy some contrivance which shall fulfil the same purpose as the accumulators do in an electrical installation4 ,even then1 however1 the financial consideration remains1 as to whether the cost of building the dam and maintaining the tide-mill in good order will not on the whole exceed the original price and the charges for the maintenance of a hundred horse power steam-engine4 ,there cannot be a doubt that in this epoch of the earth's history1 so long as the price of coal is only a few shillings a ton1 the tide-mill1 even though we seem to get its power without current expense1 is vastly more expensive than a steam-engine4 ,indeed1 ,sir ,william ,thomson remarks1 that wherever a suitable tidal basin could be found1 it would be nearly as easy to reclaim the land altogether from the sea4 ,and if this were in any locality where manufactures were possible1 the commercial value of forty acres of reclaimed land would greatly exceed all the expenses attending the steam-engine4 ,but when the time comes1 as come it apparently will1 that the price of coal shall have risen to several pounds a ton1 the economical aspect of steam as compared with other prime movers will be greatly altered2 it will then no doubt be found advantageous to utilize great sources of energy1 such as ,niagara and the tides1 which it is now more prudent to let run to waste4 ,for my argument1 however1 it matters little that the tides are not constrained to do much useful work4 ,they are always doing work of some kind1 whether that be merely heating the particles of water by friction1 or vaguely transporting sand from one part of the ocean to the other4 ,useful work or useless work are alike for the purpose of my argument4 ,we know that work can never be done unless by the consumption or transformation of energy4 ,for each unit of work that is done--whether by any machine or contrivance1 by the muscles of man or any other animal1 by the winds1 the waves1 or the tides1 or in any other way whatever--a certain equivalent quantity of energy must have been expended4 ,when1 therefore1 we see any work being performed1 we may always look for the source of energy to which the machine owes its efficiency4 ,in fact1 it is the old story illustrated1 that perpetual motion is impossible4 ,a mechanical device1 however ingenious may be the construction1 or however accurate the workmanship1 can never possess what is called perpetual motion4 ,it is needless to enter into details of any proposed contrivance of wheels1 of pumps1 of pulleys2 it is sufficient to say that nothing in the shape of mechanism can work without friction1 that friction produces heat1 that heat is a form of energy1 and that to replace the energy consumed in producing the heat there must be some source from which the machine is replenished if its motion is to be continued indefinitely4 ,hence1 as the tides may be regarded as a machine doing work1 we have to ascertain the origin of that energy which they are continually expending4 ,it is at this point that we first begin to feel the difficulties inherent in the theory of tidal evolution4 ,i do not mean difficulties in the sense of doubts1 for up to the present ,i have mentioned no doubtful point4 ,when ,i come to such ,i shall give due warning4 ,by difficulties ,i now mean points which it is not easy to understand without a little dynamical theory2 but we must face these difficulties1 and endeavour to elucidate them as well as we can4 ,let us first see what the sources of energy can possibly be on which the tides are permitted to draw4 ,our course is simplified by the fact that the energy of which we have to speak is of a mechanical description1 that is to say1 not involving heat or other more obscure forms of energy4 ,a simple type of energy is that possessed by a clock-weight after the clock has been wound4 ,a store of power is thus laid up which is gradually doled out during the week in small quantities1 second by second1 to sustain the motion of the pendulum4 ,the energy in this case is due to the fact that the weight is attracted by the earth1 and is yielded according as the weight sinks downwards4 ,in the separation between two mutually attracting bodies1 a store of energy is thus implied4 ,what we learn from an ordinary clock may be extended to the great bodies of the universe4 ,the moon is a gigantic globe separated from our earth by a distance of #bdj1jjj miles4 ,the attraction between these two bodies always tends to bring them together4 ,no doubt the moon is not falling towards the earth as the descending clock-weight is doing4 ,we may1 in fact1 consider the moon1 so far as our present object is concerned1 to be revolving almost in a circle1 of which the earth is the centre4 ,if the moon1 however1 were to be stopped1 it would at once commence to rush down towards the earth1 whither it would arrive with an awful crash in the course of four or five days4 ,it is fortunately true that the moon does not behave thus2 but it has the ability of doing so1 and thus the mere separation between the earth and the moon involves the existence of a stupendous quantity of energy1 capable under certain conditions of undergoing transformation4 ,there is also another source of mechanical energy besides that we have just referred to4 ,a rapidly moving body possesses1 in virtue of its motion1 a store of readily available energy1 and it is easy to show that energy of this type is capable of transformation into other types4 ,think of a cannon-ball rushing through the air at a speed of a thousand feet per second2 it is capable of wreaking disaster on anything which it meets1 simply because its rapid motion is the vehicle by which the energy of the gunpowder is transferred from the gun to where the blow is to be struck4 ,had the cannon been directed vertically upwards1 then the projectile1 leaving the muzzle with the same initial velocity as before1 would soar up and up1 with gradually abating speed1 until at last it reached a turning-point1 the elevation of which would depend upon the initial velocity4 ,poised for a moment at the summit1 the cannon-ball may then be likened to the clock-weight1 for the entire energy which it possessed by its motion has been transformed into the statical energy of a raised weight4 ,thus we see these two forms of energy are mutually interchangeable4 ,the raised weight if allowed to fall will acquire velocity1 or the rapidly moving weight if directed upwards will acquire altitude4 ,the quantity of energy which can be conveyed by a rapidly moving body increases greatly with its speed4 ,for instance1 if the speed of the body be doubled1 the energy will be increased fourfold1 or1 in general1 the energy which a moving body possesses may be said to be proportional to the square of its speed4 ,here then we have another source of the energy present in our earth-moon system2 for the moon is hurrying along in its path with a speed of two-thirds of a mile per second1 or about twice or three times the speed of a cannon-shot4 ,hence the fact that the moon is continuously revolving in a circle shows us that it possesses a store of energy which is nine times as great as that which a cannon-ball as massive as the moon1 and fired with the ordinary velocity1 would receive from the powder which discharged it4 ,thus we see that the moon is endowed with two sources of energy1 one of which is due to its separation from the earth1 and the other to the speed of its motion4 ,though these are distinct1 they are connected together by a link which it is important for us to comprehend4 ,the speed with which the moon revolves around the earth is connected with the moon's distance from the earth4 ,the moon might1 for instance1 revolve in a larger circle than that which it actually pursues2 but if it did so1 the speed of its motion would have to be appropriately lessened4 ,the orbit of the moon might have a much smaller radius than it has at present1 provided that the speed was sufficiently increased to compensate for the increased attraction which the earth would exercise at the lessened distance4 ,indeed1 ,i am here only stating what every one is familiar with under the form of ,kepler's ,law1 that the square of the periodic time is in proportion to the cube of the mean distance4 ,to each distance of the moon therefore belongs an appropriate speed4 ,the energy due to the moon's position and the energy due to its motion are therefore connected together4 ,one of these quantities cannot be altered without the other undergoing change4 ,if the moon's orbit were increased there would be a gain of energy due to the enlarged distance1 and a loss of energy due to the diminished speed4 ,these would not1 however1 exactly compensate4 ,on the whole1 we may represent the total energy of the moon as a single quantity1 which increases when the distance of the moon from the earth increases1 and lessens when the distance from the earth to the moon lessens4 ,for simplicity we may speak of this as moon-energy4 ,but the most important constituent of the store of energy in the earth-moon system is that contributed by the earth itself4 ,i do not now speak of the energy due to the velocity of the earth in its orbit round the sun4 ,the moon indeed participates in this equally with the earth1 but it does not affect those mutual actions between the earth and moon with which we are at present concerned4 ,we are1 in fact1 discussing the action of that piece of machinery the earth-moon system2 and its action is not affected by the circumstance that the entire machine is being bodily transported around the sun in a great annual voyage4 ,this has little more to do with the action of our present argument than has the fact that a man is walking about to do with the motions of the works of the watch in his pocket4 ,we shall1 however1 have to allude to this subject further on4 ,the energy of the earth which is significant in the earth-moon theory is due to the earth's rotation upon its axis4 ,we may here again use as an illustration the action of machinery2 and the special contrivance that ,i now refer to is the punching-engine that is used in our ship-building works4 ,in preparing a plate of iron to be riveted to the side of a ship1 a number of holes have to be made all round the margin of the plate4 ,these holes must be half an inch or more in diameter1 and the plate is sometimes as much as1 or more than1 half an inch in thickness4 ,the holes are produced in the metal by forcing a steel punch through it2 and this is accomplished without even heating the plate so as to soften the iron4 ,it is needless to say that an intense force must be applied to the punch4 ,on the other hand1 the distance through which the punch has to be moved is comparatively small4 ,the punch is attached to the end of a powerful lever1 the other end of the lever is raised by a cam1 so as to depress the punch to do its work4 ,an essential part of the machine is a small but heavy fly-wheel connected by gearing with the cam4 ,this fly-wheel when rapidly revolving contains within it1 in virtue of its motion1 a large store of energy which has gradually accumulated during the time that the punch is not actually in action4 ,the energy is no doubt originally supplied from a steam-engine4 ,what we are especially concerned with is the action of the rapidly rotating wheel as a reservoir in which a large store of energy can be conveniently maintained until such time as it is wanted4 ,in the action of punching1 when the steel die comes down upon the surface of the plate1 a large quantity of energy is suddenly demanded to force the punch against the intense resistance it experiences2 the energy for this purpose is drawn from the store in the fly-wheel1 which experiences no doubt a check in its velocity1 to be regained again from the energy of the engine during the interval which elapses before the punch is called on to make the next hole4 ,another illustration of the fly-wheel on a splendid scale is seen in our mighty steel works1 where ponderous rails are being manufactured4 ,a white-hot ingot of steel is presented to a pair of powerful rollers1 which grip the steel1 and send it through at the other side both compressed and elongated4 ,tremendous power is required to meet the sudden demand on the machine at the critical moment4 ,to obtain this power an engine of stupendous proportions is sometimes attached directly to the rollers1 but more frequently an engine of rather less horse-power will be used1 the might of this engine being applied to giving rapid rotation to an immense fly-wheel1 which may thus be regarded as a reservoir full of energy4 ,the rolling mills then obtain from this store in the fly-wheel whatever energy is necessary for their gigantic task4 ,these illustrations will suffice to show how a rapidly rotating body may contain energy in virtue of its rotation1 just as a cannon-ball contains energy in virtue of its speed of translation1 or as a clock-weight has energy in virtue of the fact that it has some distance to fall before it reaches the earth4 ,the rotating body need not necessarily have the shape of a wheel--it may be globular in form2 nor need the axes of rotation be fixed in bearings1 like those of the fly-wheel2 nor of course is there any limit to the dimensions which the rotating body may assume4 ,our earth is1 in fact1 a vast rotating body #hjjj miles in diameter1 and turning round upon its axis once every twenty-three hours and fifty-six minutes4 ,viewed in this way1 the earth is to be regarded as a gigantic fly-wheel containing a quantity of energy great in correspondence with the earth's mass4 ,the amount of energy which can be stored by rotation also depends upon the square of the velocity with which the body turns round2 thus if our earth turned round in half the time which it does at present1 that is1 if the day was twelve hours instead of twenty-four hours1 the energy contained in virtue of that rotation would be four times its present amount4 ,reverting now to the earth-moon system1 the energy which that system contains consists essentially of two parts--the moon-energy1 whose composite character ,i have already explained1 and the earth-energy1 which has its origin solely in the rotation of the earth on its axis4 ,it is necessary to observe that these are essentially distinct--there is no necessary relation between the speed of the earth's rotation and the distance of the moon1 such as there is between the distance of the moon and the speed with which it revolves in its orbit4 ,for completeness1 it ought to be added that there is also some energy due to the moon's rotation on its axis1 but this is very small for two reasons3 first1 because the moon is small compared with the earth1 and second1 because the angular velocity of the moon is also very small compared with that of the earth4 ,we may therefore dismiss as insignificant the contributions from this source of energy to the sum total4 ,i have frequently used illustrations derived from machinery1 but ,i want now to emphasize the profound distinction that exists between the rotation of the earth and the rotation of a fly-wheel in a machine shop4 ,they are both1 no doubt1 energy-holders1 but it must be borne in mind1 that as the fly-wheel doles out its energy to supply the wants of the machines with which it is connected1 a restitution of its store is continually going on by the action of the engine1 so that on the whole the speed of the fly-wheel does not slacken4 ,the earth1 however1 must be likened to a fly-wheel which has been disconnected with the engine4 ,if1 therefore1 the earth have to supply certain demands on its accumulation of energy1 it can only do so by a diminution of its hoard1 and this involves a sacrifice of some of its speed4 ,in the earth-moon system there is no engine at hand to restore the losses of energy which are inevitable when work has to be done4 ,but we have seen that work is done2 we have shown1 in fact1 that the tides are at present doing work1 and have been doing work for as long a period in the past as our imagination can extend to4 ,the energy which this work has necessitated can only have been drawn from the existing store in the system2 that energy consists of two parts--the moon-energy and the earth's rotation energy4 ,the problem therefore for us to consider is1 which of these two banks the tides have drawn on to meet their constant expenditure4 ,this is not a question that can be decided offhand2 in fact1 if we attempt to decide it in an offhand manner we shall certainly go wrong4 ,it seems so very plausible to say that as the moon causes the tides1 therefore the energy which these tides expend should be contributed by the moon4 ,but this is not the case4 ,it actually happens that though the moon does cause the tides1 yet when those tides consume energy they draw it not from the distant moon1 but from the vast supply which they find ready to their hand1 stored up in the rotation of the earth4 ,the demonstration of this is not a very simple matter2 in fact1 it is so far from being simple that many philosophers1 including some eminent ones too1 while admitting that of course the tides must have drawn their energy from one or other or both of these two sources1 yet found themselves unable to assign how the demand was distributed between the two conceivable sources of supply4 ,we are indebted to ,professor ,purser of ,belfast for having indicated the true dynamical principle on which the problem depends4 ,it involves reasoning based simply on the laws of motion and on elementary mathematics1 but not in the least involving questions of astronomical observation4 ,it would be impossible for me in a lecture like this to give any explanation of the mathematical principles referred to4 ,i shall1 however1 endeavour by some illustrations to set before you what this profound principle really is4 ,were ,i to give it the old name ,i should call it the law of the conservation of areas2 the more modern writers1 however1 speak of it as the conservation of moment of momentum1 an expression which exhibits the nature of the principle in a more definite manner4 ,i do not see how to give any very accurate illustration of what this law means1 but ,i must make the attempt1 and if you think the illustration beneath the dignity of the subject1 ,i can only plead the difficulty of mathematics as an excuse4 ,let us suppose that a ball-room is fairly filled with dancers1 or those willing to dance1 and that a merry waltz is being played2 the couples have formed1 and the floor is occupied with pairs who are whirling round and round in that delightful amusement4 ,some couples drop out for a while and others strike in2 the fewer couples there are the wider is the range around which they can waltz1 the more numerous the couples the less individual range will they possess4 ,i want you to realize that in the progress of the dance there is a certain total quantity of spin at any moment in progress2 this spin is partly made up of the rotation by which each dancer revolves round his partner1 and partly of the circular orbit about the room which each couple endeavours to describe4 ,if there are too many couples on the floor for the general enjoyment of the dance1 then both the orbit and the angular velocity of each couple will be restricted by the interference with their neighbours4 ,we may1 however1 assert that so long as the dance is in full swing the total quantity of spin1 partly rotational and partly orbital1 will remain constant4 ,when there are but few couples the unimpeded rotation and the large orbits will produce as much spin as when there is a much larger number of couples1 for in the latter case the diminished freedom will lessen the quantity of spin produced by each individual pair4 ,it will sometimes happen too that collision will take place1 but the slight diversions thus arising only increase the general merriment1 so that the total quantity of spin may be sustained1 even though one or two couples are placed temporarily hors de combat4 ,i have invoked a ball-room for the purpose of bringing out what we may call the law of the conservation of spin4 ,no matter how much the individual performers may change1 or no matter what vicissitudes arise from their collision and other mutual actions1 yet the total quantity of spin remains unchanged4 ,let us look at the earth-moon system4 ,the law of the conservation of moment of momentum may1 with sufficient accuracy for our present purpose1 be interpreted to mean that the total quantity of spin in the system remains unaltered4 ,in our system the spin is threefold2 there is first the rotation of the earth on its axis1 there is the rotation of the moon on its axis1 and then there is the orbital revolution of the moon around the earth4 ,the law to which we refer asserts that the total quantity of these three spins1 each estimated in the proper way1 will remain constant4 ,it matters not that tides may ebb and flow1 or that the distribution of the spin shall vary1 but its total amount remains inflexibly constant4 ,one constituent of the total amount--that is1 the rotation of the moon on its axis--is so insignificant1 that for our present purposes it may be entirely disregarded4 ,we may therefore assert that the amount of spin in the earth1 due to its rotation round its axis1 added to the amount of spin in the moon due to its revolution round the earth1 remains unalterable4 ,if one of these quantities change by increase or by decrease1 the other must correspondingly change by decrease or by increase4 ,if1 therefore1 from any cause1 the earth began to spin a little more quickly round its axis1 the moon must do a little less spin2 and consequently1 it must shorten its distance from the earth4 ,or suppose that the earth's velocity of rotation is abated1 then its contribution to the total amount of spin is lessened2 the deficiency must therefore be made up by the moon1 but this can only be done by an enlargement of the moon's orbit4 ,i should add1 as a caution1 that these results are true only on the supposition that the earth-moon system is isolated from all external interference4 ,with this proviso1 however1 it matters not what may happen to the earth or moon1 or what influence one of them may exert upon the other1 no matter what tides may be raised1 no matter even if the earth fly into fragments1 the whole quantity of spin of all those fragments would1 if added to the spin of the moon1 yield the same unalterable total4 ,we are here in possession of a most valuable dynamical principle4 ,we are not concerned with any special theory as to the action of the tides2 it is sufficient for us that in some way or other the tides have been caused by the moon1 and that being so1 the principle of the conservation of spin will apply4 ,were the earth and the moon both rigid bodies1 then there could be of course no tides on the earth1 it being rigid and devoid of ocean4 ,the rotation of the earth on its axis would therefore be absolutely without change1 and therefore the necessary condition of the conservation of spin would be very simply attained by the fact that neither of the constituent parts changed4 ,the earth1 however1 not being entirely rigid1 and being subject to tides1 this simple state of things cannot continue2 there must be some change in progress4 ,i have already shown that the fact of the ebbing and the flowing of the tide necessitates an expenditure of energy1 and we saw that this energy must come either from that stored up in the earth by its rotation1 or from that possessed by the moon in virtue of its distance and revolution4 ,the law of the conservation of spin will enable us to decide at once as to whence the tides get their energy4 ,suppose they took it from the moon1 the moon would then lose in energy1 and consequently come nearer the earth4 ,the quantity of spin contributed by the moon would therefore be lessened1 and accordingly the spin to be made up by the earth would be increased4 ,that means1 of course1 that the velocity of the earth rotating on its axis must be increased1 and this again would necessitate an increase in the earth's rotational energy4 ,it can be shown1 too1 that to keep the total spin right1 the energy of the earth would have to gain more than the moon would have lost by revolving in a smaller orbit4 ,thus we find that the total quantity of energy in the system would be increased4 ,this would lead to the absurd result that the action of the tides manufactured energy in our system4 ,of course1 such a doctrine cannot be true2 it would amount to a perpetual motion6 ,we might as well try to get a steam-engine which would produce enough heat by friction not only to supply its own boilers1 but to satisfy all the thermal wants of the whole parish4 ,we must therefore adopt the other alternative4 ,the tides do not draw their energy from the moon2 they draw it from the store possessed by the earth in virtue of its rotation4 ,we can now state the end of this rather long discussion in a very simple and brief manner4 ,energy can only be yielded by the earth at the expense of some of the speed of its rotation4 ,the tides must therefore cause the earth to revolve more slowly2 in other words1 the tides are increasing the length of the day4 ,the earth therefore loses some of its velocity of rotation2 consequently it does less than its due share of the total quantity of spin1 and an increased quantity of spin must therefore be accomplished by the moon2 but this can only be done by an enlargement of its orbit4 ,thus there are two great consequences of the tides in the earth-moon system--the days are getting longer1 the moon is receding further4 ,these points are so important that ,i shall try and illustrate them in another way1 which will show1 at all events1 that one and both of these tidal phenomena commend themselves to our common sense4 ,have we not shown how the tides in their ebb and flow are incessantly producing friction1 and have we not also likened the earth to a great wheel8 ,when the driver wants to stop a railway train the brakes are put on1 and the brake is merely a contrivance for applying friction to the circumference of a wheel for the purpose of checking its motion4 ,or when a great weight is being lowered by a crane1 the motion is checked by a band which applies friction on the circumference of a wheel1 arranged for the special purpose4 ,need we then be surprised that the friction of the tides acts like a brake on the earth1 and gradually tends to check its mighty rotation8 ,the progress of lengthening the day by the tides is thus readily intelligible4 ,it is not quite so easy to see why the ebbing and the flowing of the tide on the earth should actually have the effect of making the moon to retreat2 this phenomenon is in deference to a profound law of nature1 which tells us that action and reaction are equal and opposite to each other4 ,if ,i might venture on a very homely illustration1 ,i may say that the moon1 like a troublesome fellow1 is constantly annoying the earth by dragging its waters backward and forward by means of tides2 and the earth1 to free itself from this irritating interference1 tries to push off the aggressor and to make him move further away4 ,illustration3 ,,fig4 #b4 ,another way in which we can illustrate the retreat of the moon as the inevitable consequence of tidal friction is shown in the adjoining figure1 in which the large body ,e represents the earth1 and the small body ,m the moon4 ,we may for simplicity regard the moon as a point1 and as this attracts each particle of the earth1 the total effect of the moon on the earth may be represented by a single force4 ,by the law of equality of action and reaction1 the force of the earth on the moon is to be represented by an equal and opposite force4 ,if there were no tides then the moon's force would of course pass through the earth's centre2 but as the effect of the moon is to slacken the earth's rotation1 it follows that the total force does not exactly pass through the line of the earth's centre1 but a little to one side1 in order to pull the opposite way to that in which the earth is turning1 and thus bring down its speed4 ,we may therefore decompose the earth's total force on the moon into two parts1 one of which tends directly towards the earth's centre1 while the other acts tangentially to the moon's orbit4 ,the central force is of course the main guiding power which keeps the moon in its path2 but the incessant tangential force constantly tends to send the moon out further and further1 and thus the growth of its orbit can be accounted for4 ,we therefore conclude finally1 that the tides are making the day longer and sending the moon away further4 ,it is the development of the consequences of these laws that specially demands our attention in these lectures4 ,we must have the courage to look at the facts unflinchingly1 and deduce from them all the wondrous consequences they involve4 ,their potency arises from a characteristic feature--they are unintermitting4 ,most of the great astronomical changes with which we are ordinarily familiar are really periodic3 they gradually increase in one direction for years1 for centuries1 or for untold ages2 but then a change comes1 and the increase is changed into a decrease1 so that after the lapse of becoming periods the original state of things is restored4 ,such periodic phenomena abound in astronomy4 ,there is the annual fluctuation of the seasons2 there is the eighteen or nineteen year period of the moon2 there is the great period of the precession of the equinoxes1 amounting to twenty-six thousand years2 and then there is the stupendous ,annus ,magnus of hundreds of thousands of years1 during which the earth's orbit itself breathes in and out in response to the attraction of the planets4 ,but these periodic phenomena1 however important they may be to us mere creatures of a day1 are insignificant in their effects on the grand evolution through which the celestial bodies are passing4 ,the really potent agents in fashioning the universe are those which1 however slow or feeble they may seem to be1 are still incessant in their action4 ,the effect which a cause shall be competent to produce depends not alone upon the intensity of that cause1 but also upon the time during which it has been in operation4 ,from the phenomena of geology1 as well as from those of astronomy1 we know that this earth and the system to which it belongs has endured for ages1 not to be counted by scores of thousands of years1 or1 as ,prof4 ,tyndall has so well remarked1 ,not for six thousand years1 nor for sixty thousand years1 nor six hundred thousand years1 but for ons of untold millions4 ,those slender agents which have devoted themselves unceasingly to the accomplishment of a single task may in this long lapse of time have accomplished results of stupendous magnitude4 ,in famed stalactite caverns we are shown a colossal figure of crystal extending from floor to roof1 and the formation of that column is accounted for when we see a tiny drop falling from the roof above to the floor beneath4 ,a lifetime may not suffice for that falling drop to add an appreciable increase to the stalactite down which it trickles1 or to the growing stalagmite on which it falls2 but when the operation has been in progress for immense ages1 it is capable of the formation of the stately column4 ,here we have an illustration of an influence which1 though apparently trivial1 acquires colossal significance when adequate time is afforded4 ,it is phenomena of this kind which the student of nature should most narrowly watch1 for they are the real architects of the universe4 ,the tidal consequences which we have already demonstrated are emphatically of this non-periodic class--the day is always lengthening1 the moon is always retreating4 ,to-day is longer than yesterday2 to-morrow will be longer than to-day4 ,it cannot be said that the change is a great one2 it is indeed too small to be appreciable even by our most delicate observations4 ,in one thousand years the alteration in the length of a day is only a small fraction of a second2 but what may be a very small matter in one thousand years can become a very large one in many millions of years4 ,thus it is that when we stretch our view through immense vistas of time past1 or when we look forward through immeasurable ages of time to come1 the alteration in the length of the day will assume the most startling proportions1 and involve the most momentous consequences4 ,let us first look back4 ,there was a time when the day1 instead of being the twenty-four hours we now have1 must have been only twenty-three hours1 ,how many millions of years ago that was ,i do not pretend to say1 nor is the point material for our argument2 suffice it to say1 that assuming1 as geology assures us we may assume1 the existence of these ons of millions of years1 there was once a time when the day was not only one hour shorter1 but was even several hours less than it is at present4 ,nor need we stop our retrospect at a day of even twenty1 or fifteen1 or ten hours long2 we shall at once project our glance back to an immeasurably remote epoch1 at which the earth was spinning round in a time only one sixth or even less of the length of the present day4 ,there is here a reason for our retrospect to halt1 for at some eventful period1 when the day was about three or four hours long1 the earth must have been in a condition of a very critical kind4 ,it is well known that fearful accidents occasionally happen where large grindstones are being driven at a high speed4 ,the velocity of rotation becomes too great for the tenacity of the stone to withstand the stress2 a rupture takes place1 the stone flies in pieces1 and huge fragments are hurled around4 ,for each particular grindstone there is a certain special velocity depending upon its actual materials and character1 at which it would inevitably fly in pieces4 ,i have once before likened our earth to a wheel2 now let me liken it to a grindstone4 ,there is therefore a certain critical velocity of rotation for the earth at which it would be on the brink of rupture4 ,we cannot exactly say1 in our ignorance of the internal constitution of the earth1 what length of day would be the shortest possible for our earth to have consistently with the preservation of its integrity2 we may1 however1 assume that it will be about three or four hours1 or perhaps a little less than three4 ,the exact amount1 however1 is not really very material to us2 it would be sufficient for our argument to assert that there is a certain minimum length of day for which the earth can hold together4 ,in our retrospect1 therefore1 through the abyss of time past our view must be bounded by that state of the earth when it is revolving in this critical period4 ,with what happened before that we shall not at present concern ourselves4 ,thus we look back to a time at the beginning of the present order of things1 when the day was only some three or four hours long4 ,let us now look at the moon1 and examine where it must have been during these past ages4 ,as the moon is gradually getting further and further from us at present1 so1 looking back into past time1 we find that the moon was nearer and nearer to the earth the further back our view extends2 in fact1 concentrating our attention solely on essential features1 we may say that the path of the moon is a sort of spiral which winds round and round the earth1 gradually getting larger1 though with extreme slowness4 ,looking back we see this spiral gradually coiling in and in1 until in a retrospect of millions of years1 instead of its distance from the earth being #bdj1jjj miles1 it must have been much less4 ,there was a time when the moon was only #bjj1jjj miles away2 there was a time many millions of years ago1 when the moon was only #ajj1jjj miles away4 ,nor can we here stop our retrospect2 we must look further and further back1 and follow the moon's spiral path as it creeps in and in towards the earth1 until at last it appears actually in contact with that great globe of ours1 from which it is now separated by a quarter of a million of miles4 ,surely the tides have thus led us to the knowledge of an astounding epoch in our earth's past history1 when the earth is spinning round in a few hours1 and when the moon is1 practically speaking1 in contact with it4 ,perhaps ,i should rather say1 that the materials of our present moon were in this situation1 for we would hardly be entitled to assume that the moon then possessed the same globular form in which we see it now4 ,to form a just apprehension of the true nature of both bodies at this critical epoch1 we must study their concurrent history as it is disclosed to us by a totally different line of reasoning4 ,drop1 then1 for a moment all thought of tides1 and let us bring to our aid the laws of heat1 which will disclose certain facts in the ancient history of the earth-moon system perhaps as astounding as those to which the tides have conducted us4 ,in one respect we may compare these laws of heat with the laws of the tides2 they are both alike non-periodic1 their effects are cumulative from age to age1 and imagination can hardly even impose a limit to the magnificence of the works they can accomplish4 ,our argument from heat is founded on a very simple matter4 ,it is quite obvious that a heated body tends to grow cold4 ,i am not now speaking of fires or of actual combustion whereby heat is produced2 ,i am speaking merely of such heat as would be possessed by a red-hot poker after being taken from the fire1 or by an iron casting after the metal has been run into the mould4 ,in such cases as this the general law holds good1 that the heated body tends to grow cold4 ,the cooling may be retarded no doubt if the passage of heat from the body is impeded4 ,we can1 for instance1 retard the cooling of a teapot by the well-known practice of putting a cosy upon it2 but the law remains that1 slowly or quickly1 the heated body will tend to grow colder4 ,it seems almost puerile to insist with any emphasis on a point so obvious as this1 but yet ,i frequently find that people do not readily apprehend all the gigantic consequences that can flow from a principle so simple4 ,it is true that a poker cools when taken from the fire2 we also find that a gigantic casting weighing many tons will grow gradually cold1 though it may require days to do so4 ,the same principle will extend to any object1 no matter how vast it may happen to be4 ,were that great casting #bjjj miles in diameter1 or were it #hjjj miles in diameter1 it will still steadily part with its heat1 though no doubt the process of cooling becomes greatly prolonged with an increase in the dimensions of the heated body4 ,the earth and the moon cannot escape from the application of these simple principles4 ,let us first speak of the earth4 ,there are multitudes of volcanoes in action at the present moment in various countries upon this earth4 ,now whatever explanation may be given of the approximate cause of the volcanic phenomena1 there can be no doubt that they indicate the existence of heat in the interior of the earth4 ,it may possibly be1 as some have urged1 that the volcanoes are merely vents for comparatively small masses of subterranean molten matter2 it may be1 as others more reasonably1 in my opinion1 believe1 that the whole interior of the earth is at the temperature of incandescence1 and that the eruptions of volcanoes and the shocks of earthquakes are merely consequences of the gradual shrinkage of the external crust1 as it continually strives to accommodate itself to the lessening bulk of the fluid interior4 ,but whichever view we may adopt1 it is at least obvious that the earth is in part1 at all events1 a heated body1 and that the heat is not in the nature of a combustion1 generated and sustained by the progress of chemical action4 ,no doubt there may be local phenomena of this description1 but by far the larger proportion of the earth's internal heat seems merely the fervour of incandescence4 ,it is to be likened to the heat of the molten iron which has been run into the sand1 rather than to the glowing coals in the furnace in which that iron has been smelted4 ,there is one volcanic outbreak of such exceptional interest in these modern times that ,i cannot refrain from alluding to it4 ,doubtless every one has heard of that marvellous eruption of ,krakatoa1 which occurred on ,august #bfth and #bgth1 #ahhc1 and gives a unique chapter in the history of volcanic phenomena4 ,not alone was the eruption of ,krakatoa alarming in its more ordinary manifestations1 but it was unparalleled both in the vehemence of the shock and in the distance to which the effects of the great eruption were propagated4 ,i speak not now of the great waves of ocean that inundated the coasts of ,sumatra and ,java1 and swept away thirty-six thousand people1 nor do ,i allude to the intense darkness which spread for one hundred and eighty miles or more all round4 ,i shall just mention the three most important phenomena1 which demonstrate the energy which still resides in the interior of our earth4 ,place a terrestrial globe before you1 and fix your attention on the ,straits of ,sunda2 think also of the great atmospheric ocean some two or three hundred miles deep which envelopes our earth4 ,when a pebble is tossed into a pond a beautiful series of concentric ripples diverge from it2 so when ,krakatoa burst up in that mighty catastrophe1 a series of gigantic waves were propagated through the air2 they embraced the whole globe1 converged to the antipodes of ,krakatoa1 thence again diverged1 and returned to the seat of the volcano2 a second time the mighty series of atmospheric ripples spread to the antipodes1 and a second time returned4 ,seven times did that series of waves course over our globe1 and leave their traces on every self-recording barometer that our earth possesses4 ,thirty-six hours were occupied in the journey of the great undulation from ,krakatoa to its antipodes4 ,perhaps even more striking was the extent of our earth's surface over which the noise of the great explosion spread4 ,at ,batavia1 ninety-four miles away1 the concussions were simply deafening2 at ,macassar1 in ,celebes1 two steamers were sent out to investigate the explosions which were heard1 little thinking that they came from ,krakatoa1 nine hundred and sixty-nine miles away4 ,alarming sounds were heard over the island of ,timor1 one thousand three hundred and fifty-one miles away from ,krakatoa4 ,diego ,garcia in the ,chogos islands is two thousand two hundred and sixty-seven miles from ,krakatoa1 but the thunders traversed even this distance1 and were attributed to some ship in distress1 for which a search was made4 ,most astounding of all1 there is undoubted evidence that the sound of the mighty explosion was propagated across nearly the entire ,indian ocean1 and was heard in the island of ,rodriguez1 almost three thousand miles away4 ,the immense distance over which this sound journeyed will be appreciated by the fact1 that the noise did not reach ,rodriguez until four hours after it had left ,krakatoa4 ,in fact1 it would seem that if ,vesuvius were to explode with the same vehemence as ,krakatoa did1 the thunders of the explosion might penetrate so far as to be heard in ,london4 ,there is another and more beautiful manifestation of the world-wide significance of the ,krakatoa outbreak4 ,the vast column of smoke and ashes ascended twenty miles high in the air1 and commenced a series of voyages around the equatorial regions of the earth4 ,in three days it crossed the ,indian ocean1 and was traversing equatorial ,africa2 then came an ,atlantic voyage2 and then it coursed over central ,america1 before a ,pacific voyage brought it back to its point of departure after thirteen days2 then the dust started again1 and was traced around another similar circuit1 while it was even tracked for a considerable time in placing the third girdle round the earth4 ,strange blue suns and green moons and other mysterious phenomena marked the progress of this vast volcanic cloud4 ,at last the cloud began to lose its density1 the dust spread more widely over the tropics1 became diffused through the temperate regions1 and then the whole earth was able to participate in the glories of ,krakatoa4 ,the marvellous sunsets in the autumn of #ahhc are attributable to this cause2 and thus once again was brought before us the fact that the earth still contains large stores of thermal energy4 ,attempts are sometimes made to explain volcanic phenomena on the supposition that they are entirely of a local character1 and that we are not entitled to infer the incandescent nature of the earth's interior from the fact that volcanic outbreaks occasionally happen4 ,for our present purpose this point is immaterial1 though ,i must say it appears to me unreasonable to deny that the interior of the earth is in a most highly heated state4 ,every test we can apply shows us the existence of internal heat4 ,setting aside the more colossal phenomena of volcanic eruptions1 we have innumerable minor manifestations of its presence4 ,are there not geysers and hot springs in many parts of the earth8 and have we not all over our globe invariable testimony confirming the statement1 that the deeper we go down beneath its surface the hotter does the temperature become8 ,every miner is familiar with these facts2 he knows that the deeper are his shafts the warmer it is down below1 and the greater the necessity for providing increased ventilation to keep the temperature within a limit that shall be suitable for the workmen4 ,all these varied classes of phenomena admit solely of one explanation1 and that is1 that the interior of the earth contains vast stores of incandescent heat4 ,we now apply to our earth the same reasoning which we should employ on a poker taken from the fire1 or on a casting drawn from the foundry4 ,such bodies will lose their heat by radiation and conduction4 ,the earth is therefore losing its heat4 ,no doubt the process is an extremely slow one4 ,the mighty reservoirs of internal heat are covered by vast layers of rock1 which are such excellent non-conductors that they offer every possible impediment to the leakage of heat from the interior to the surface4 ,we coat our steam-pipes over with non-conducting material1 and this can now be done so successfully1 that it is beginning to be found economical to transmit steam for a very long distance through properly protected pipes4 ,but no non-conducting material that we can manufacture can be half so effective as the shell of rock twenty miles or more in thickness1 which secures the heated interior of the earth from rapid loss by radiation into space4 ,even were the earth's surface solid copper or solid silver1 both most admirable conductors of heat1 the cooling down of this vast globe would be an extremely tardy process2 how much more tardy must it therefore be when such exceedingly bad conductors as rocks form the envelope8 ,how imperfectly material of this kind will transmit heat is strikingly illustrated by the great blast iron furnaces which are so vitally important in one of ,england's greatest manufacturing industries4 ,a glowing mass of coal and iron ore and limestone is here urged to vivid incandescence by a blast of air itself heated to an intense temperature4 ,the mighty heat thus generated--sufficient as it is to detach the iron from its close alliance with the earthy materials and to render the metal out as a pure stream rushing white-hot from the vent--is sufficiently confined by a few feet of brick-work1 one side of which is therefore at the temperature of molten iron1 while the other is at a temperature not much exceeding that of the air4 ,we may liken the brick-work of a blast furnace to the rocky covering of the earth2 in each case an exceedingly high temperature on one side is compatible with a very moderate temperature on the other4 ,although the drainage of heat away from the earth's interior to its surface1 and its loss there by radiation into space1 is an extremely tardy process1 yet it is incessantly going on4 ,we have here again to note the ability for gigantic effect which a small but continually operating cause may have1 provided it always tends in the same direction4 ,the earth is incessantly losing heat2 and though in a day1 a week1 or a year the loss may not be very significant1 yet when we come to deal with periods of time that have to be reckoned by millions of years1 it may well be that the effect of a small loss of heat per annum can1 in the course of these ages1 reach unimagined dimensions4 ,suppose1 for instance1 that the earth experienced a fall of temperature in its interior which amounted to only one-thousandth of a degree in a year4 ,so minute a quantity as this is imperceptible4 ,even in a century1 the loss of heat at this rate would be only the tenth of a degree4 ,there would be no possible way of detecting it2 the most careful thermometer could not be relied on to tell us for a certainty that the temperature of the hot waters of ,bath had declined the tenth of a degree2 and ,i need hardly say1 that the fall of a tenth of a degree would signify nothing in the lavas of ,vesuvius1 nor influence the thunders of ,krakatoa by one appreciable note4 ,so far as a human life or the life of the human race is concerned1 the decline of a tenth of a degree per century in the earth's internal heat is absolutely void of significance4 ,i cannot1 however1 impress upon you too strongly1 that the mere few thousands of years with which human history is cognizant are an inappreciable moment in comparison with those unmeasured millions of years which geology opens out to us1 or with those far more majestic periods which the astronomer demands for the events he has to describe4 ,an annual loss of even one-thousandth of a degree will be capable of stupendous achievements when supposed to operate during epochs of geological magnitude4 ,in fact1 its effects would be so vast1 that it seems hardly credible that the present loss of heat from the earth should be so great as to amount to an abatement of one-thousandth of a degree per annum1 for that would mean1 that in a thousand years the earth's temperature would decline by one degree1 and in a million years the decline would amount to a thousand degrees4 ,at all events1 the illustration may suffice to show1 that the fact that we are not able to prove by our instruments that the earth is cooling is no argument whatever against the inevitable law1 that the earth1 like every other heated body1 must be tending towards a lower temperature4 ,without pretending to any numerical accuracy1 we can at all events give a qualitative if not a quantitative analysis of the past history of our earth1 in so far as its changes of temperature are concerned4 ,a million years ago our earth doubtless contained appreciably more heat than it does at present4 ,i speak not now1 of course1 of mere solar heat--of the heat which gives us the vicissitudes of seasons2 ,i am only referring to the original hoard of internal heat which is gradually waning4 ,as therefore our retrospect extends through millions and millions of past ages1 we see our earth ever growing warmer and warmer the further and further we look back4 ,there was a time when those heated strata which we have now to go deep down in mines to find were considerably nearer the surface4 ,at present1 were it not for the sun1 the heat of the earth where we stand would hardly be appreciably above the temperature of infinite space--perhaps some #bjj or #cjj degrees below zero4 ,but there must have been a time when there was sufficient internal heat to maintain the exterior at a warm and indeed at a very hot temperature4 ,nor is there any bound to our retrospect arising from the operation or intervention of any other agent1 so far as we know2 consequently the hotter and the hotter grows the surface the further and the further we look back4 ,nor can we stop until1 at an antiquity so great that ,i do not venture on any estimate of the date1 we discover that this earth must have consisted of glowing hot material4 ,further and further we can look back1 and we see the rocks--or whatever other term we choose to apply to the then ingredients of the earth's crust--in a white-hot and even in a molten condition4 ,thus our argument has led us to the belief that time was when this now solid globe of ours was a ball of white-hot fluid4 ,on the argument which ,i have here used there are just two remarks which ,i particularly wish to make4 ,note in the first place1 that our reasoning is founded on the fact that the earth is at present1 to some extent1 heated4 ,it matters not whether this heat be much or little2 our argument would have been equally valid had the earth only contained a single particle of its mass at a somewhat higher temperature than the temperature of space4 ,i am1 of course1 not alluding in this to heat which can be generated by combustion4 ,the other point to which ,i refer is to remove an objection which may possibly be urged against this line of reasoning4 ,i have argued that because the temperature is continually increasing as we look backwards1 that therefore a very great temperature must once have prevailed4 ,without some explanation this argument is not logically complete4 ,there is1 it is well known1 the old paradox of the geometric series2 you may add a farthing to a halfpenny1 and then a half-farthing1 and then a quarter-farthing1 and then the eighth of a farthing1 followed by the sixteenth1 and thirty-second1 and so on1 halving the contribution each time4 ,now no matter how long you continue this process1 even if you went on with it for ever1 and thus made an infinite number of contributions1 you would never accomplish the task of raising the original halfpenny to the dignity of a penny4 ,an infinite number of quantities may therefore1 as this illustration shows1 never succeed in attaining any considerable dimensions4 ,our argument1 however1 with regard to the increase of heat as we look back is the very opposite of this4 ,it is the essence of a cooling body to lose heat more rapidly in proportion as its temperature is greater4 ,thus though the one-thousandth of a degree may be all the fall of temperature that our earth now experiences in a twelvemonth1 yet in those glowing days when the surface was heated to incandescence1 the loss of heat per annum must have been immensely greater than it is now4 ,it therefore follows that the rate of gain of the earth's heat as we look back must be of a different character to that of the geometric series which ,i have just illustrated2 for each addition to the earth's heat1 as we look back from year to year1 must grow greater and greater1 and therefore there is here no shelter for a fallacy in the argument on which the existence of high temperature of primeval times is founded4 ,the reasoning that ,i have applied to our earth may be applied in almost similar words to the moon4 ,it is true that we have not any knowledge of the internal nature of the moon at present1 nor are we able to point to any active volcanic phenomena at present in progress there in support of the contention that the moon either has now internal heat1 or did once possess it4 ,it is1 however1 impossible to deny the evidence which the lunar craters afford as to the past existence of volcanic activity on our satellite4 ,heat1 therefore1 there was once in the moon2 and accordingly we are enabled to conclude that1 on a retrospect through illimitable periods of time1 we must find the moon transformed from that cold and inert body she now seems to a glowing and incandescent mass of molten material4 ,the earth therefore and the moon in some remote ages--not alone anterior to the existence of life1 but anterior even to the earliest periods of which geologists have cognizance--must have been both globes of molten materials which have consolidated into the rocks of the present epoch4 ,we must now revert to the tidal history of the earth-moon system4 ,did we not show that there was a time when the earth and the moon--or perhaps1 ,i should say1 the ingredients of the earth and moon--were close together1 were indeed in actual contact8 ,we have now learned1 from a wholly different line of reasoning1 that in very early ages both bodies were highly heated4 ,here as elsewhere in this theory we can make little or no attempt to give any chronology1 or to harmonize the different lines along which the course of history has run4 ,no one can form the slightest idea as to what the temperature of the earth and of the moon must have been in those primeval ages when they were in contact4 ,it is impossible1 however1 to deny that they must both have been in a very highly heated state2 and everything we know of the matter inclines us to the belief that the temperature of the earth-moon system must at this critical epoch have been one of glowing incandescence and fusion4 ,it is therefore quite possible that these bodies--the moon especially--may have then been not at all of the form we see them now4 ,it has been supposed1 and there are some grounds for the supposition1 that at this initial stage of earth-moon history the moon materials did not form a globe1 but were disposed in a ring which surrounded the earth1 the ring being in a condition of rapid rotation4 ,it was at a subsequent period1 according to this view1 that the substances in the ring gradually drew together1 and then by their mutual attractions formed a globe which ultimately consolidated down into the compact moon as we now see it4 ,i must1 however1 specially draw your attention to the clearly-marked line which divides the facts which dynamics have taught us from those notions which are to be regarded as more or less conjectural4 ,interpreting the action of the tides by the principles of dynamics1 we are assured that the moon was once--or rather the materials of the moon--in the immediate vicinity of the earth4 ,there1 however1 dynamics leaves us1 and unfortunately withholds its accurate illumination from the events which immediately preceded that state of things4 ,the theory of tidal evolution which ,i am describing in these lectures is mainly the work of ,professor ,george ,h4 ,darwin of ,cambridge4 ,much of the original parts of the theory of the tides was due to ,sir ,william ,thomson1 and ,i have also mentioned how ,professor ,purser contributed an important element to the dynamical theory4 ,it is1 however1 ,darwin who has persistently deduced from the theory all the various consequences which can be legitimately drawn from it4 ,darwin1 for instance1 pointed out that as the moon is receding from us1 it must1 if we only look far enough back1 have been once in practical contact with the earth4 ,it is to ,darwin also that we owe many of the other parts of a fascinating theory1 either in its mathematical or astronomical aspect2 but ,i must take this opportunity of saying1 that ,i do not propose to make ,professor ,darwin or any of the other mathematicians ,i have named responsible for all that ,i shall say in these lectures4 ,i must be myself accountable for the way in which the subject is being treated1 as well as for many of the illustrations used1 and some of the deductions ,i have drawn from the subject4 ,it is almost unavoidable for us to make a surmise as to the cause by which the moon had come into this remarkable position close to the earth at the most critical epoch of earth-moon history4 ,with reference to this ,professor ,darwin has offered an explanation1 which seems so exceedingly plausible that it is impossible to resist the notion that it must be correct4 ,i will ask you to think of the earth not as a solid body covered largely with ocean1 but as a glowing globe of molten material4 ,in a globe of this kind it is possible for great undulations to be set up4 ,here is a large vase of water1 and by displacing it ,i can cause the water to undulate with a period which depends on the size of the vessel2 undulations can be set up in a bucket of water1 the period of these undulations being dependent upon the dimensions of the bucket4 ,similarly in a vast globe of molten material certain undulations could be set up1 and those undulations would have a period depending upon the dimensions of this vibrating mass4 ,we may conjecture a mode in which such vibrations could be originated4 ,imagine a thin shell of rigid material which just encases the globe2 suppose this be divided into four quarters1 like the four quarters of an orange1 and that two of these opposite quarters be rejected1 leaving two quarters on the liquid4 ,now suppose that these two quarters be suddenly pressed in1 and then be as suddenly removed--they will produce depressions1 of course1 on the two opposite quarters1 while the uncompressed quarters will become protuberant4 ,in virtue of the mutual attractions between the different particles of the mass1 an effort will be made to restore the globular form1 but this will of course rather overshoot the mark2 and therefore a series of undulations will be originated by which two opposite quarters of the sphere will alternately shrink in and become protuberant4 ,there will be a particular period to this oscillation4 ,for our globe it would appear to be somewhere about an hour and a half or two hours2 but there is necessarily a good deal of uncertainty about this point4 ,we have seen how in those primitive days the earth was spinning around very rapidly2 and ,i have also stated that the earth might at this very critical epoch of its history be compared with a grindstone which is being driven so rapidly that it is on the very brink of rupture4 ,it is remarkable to note1 that a cause tending to precipitate a rupture of the earth was at hand4 ,the sun then raised tides in the earth as it does at present4 ,when the earth revolved in a period of some four hours or thereabouts1 the high tides caused by the sun succeeded each other at intervals of about two hours4 ,when ,i speak of tides in this respect1 of course ,i am not alluding to oceanic tides2 these were the days long before ocean existed1 at least in the liquid form4 ,the tides ,i am speaking about were raised in the fluids and materials which then constituted the whole of the glowing earth2 those tides rose and fell under the throb produced by the sun1 just as truly as tides produced in an ordinary ocean4 ,but now note the significant coincidence between the period of the throb produced by the sun-raised tides1 and that natural period of vibration which belonged to our earth as a mass of molten material4 ,it therefore follows1 that the impulse given to the earth by the sun harmonized in time with that period in which the earth itself was disposed to oscillate4 ,a well-known dynamical principle here comes into play4 ,you see a heavy weight hanging by a string1 and in my hand ,i hold a little slip of wood no heavier than a common pencil2 ordinarily speaking1 ,i might strike that heavy weight with this slip of wood1 and no effect is produced2 but if ,i take care to time the little blows that ,i give so that they shall harmonize with the vibrations which the weight is naturally disposed to make1 then the effect of many small blows will be cumulative1 so much so1 that after a short time the weight begins to respond to my efforts1 and now you see it has acquired a swing of very considerable amplitude4 ,in ,professor ,fitzgerald's address to the ,british ,association at ,bath last autumn1 he gives an account of those astounding experiments of ,hertz1 in which well-timed electrical impulses broke down an air resistance1 and revealed to us ethereal vibrations which could never have been made manifest except by the principle we are here discussing4 ,the ingenious conjecture has been made1 that when the earth was thrown into tidal vibrations in those primeval days1 these slight vibrations1 harmonizing as they did with the natural period of the earth1 gradually acquired amplitude2 the result being that the pulse of each successive vibration increased at last to such an extent that the earth separated under the stress1 and threw off a portion of those semi-fluid materials of which it was composed4 ,in process of time these rejected portions contracted together1 and ultimately formed that moon we now see4 ,such is the origin of the moon which the modern theory of tidal evolution has presented to our notice4 ,there are two great epochs in the evolution of the earth-moon system--two critical epochs which possess a unique dynamical significance2 one of these periods was early in the beginning1 while the other cannot arrive for countless ages yet to come4 ,i am aware that in discussing this matter ,i am entering somewhat largely into mathematical principles2 ,i must only endeavour to state the matter as succinctly as the subject will admit4 ,in an earlier part of this lecture ,i have explained how1 during all the development of the earth-moon system1 the quantity of moment of momentum remains unaltered4 ,the moment of momentum of the earth's rotation added to the moment of momentum of the moon's revolution remains constant2 if one of these quantities increase the other must decrease1 and the progress of the evolution will have this result1 that energy shall be gradually lost in consequence of the friction produced by the tides4 ,the investigation is one appropriate for mathematical formul1 such as those that can be found in ,professor ,darwin's memoirs2 but nature has in this instance dealt kindly with us1 for she has enabled an abstruse mathematical principle to be dealt with in a singularly clear and concise manner4 ,we want to obtain a definite view of the alteration in the energy of the system which shall correspond to a small change in the velocity of the earth's rotation1 the moon of course accommodating itself so that the moment of momentum shall be preserved unaltered4 ,we can use for this purpose an angular velocity which represents the excess of the earth's rotation over the angular revolution of the moon2 it is1 in fact1 the apparent angular velocity with which the moon appears to move round the heavens4 ,if we represent by ,n the angular velocity of the earth1 and by ,m the angular velocity of the moon in its orbit round the earth1 the quantity we desire to express is ,n--,m2 we shall call it the relative rotation4 ,the mathematical theorem which tells us what we want can be enunciated in a concise manner as follows4 ,the alteration of the energy of the system may be expressed by multiplying the relative rotation by the change in the earth's angular velocity4 ,this result will explain many points to us in the theory1 but just at present ,i am only going to make a single inference from it4 ,i must advert for a moment to the familiar conception of a maximum or a minimum4 ,if a magnitude be increasing1 that is1 gradually growing greater and greater1 it has obviously not attained a maximum so long as the growth is in progress4 ,nor if the object be actually decreasing can it be said to be at a maximum either2 for then it was greater a second ago than it is now1 and therefore it cannot be at a maximum at present4 ,we may illustrate this by the familiar example of a stone thrown up into the air2 at first it gradually rises1 being higher at each instant than it was previously1 until a culminating point is reached1 when just for a moment the stone is poised at the summit of its path ere it commences its return to earth again4 ,in this case the maximum point is obtained when the stone1 having ceased to ascend1 and not having yet commenced to descend1 is momentarily at rest4 ,the same principles apply to the determination of a minimum4 ,as long as the magnitude is declining the minimum has not been reached2 it is only when the decline has ceased1 and an increase is on the point of setting in1 that the minimum can be said to be touched4 ,the earth-moon system contains at any moment a certain store of energy1 and to every conceivable condition of the earth-moon system a certain quantity of energy is appropriate4 ,it is instructive for us to study the different positions in which the earth and the moon might lie1 and to examine the different quantities of energy which the system will contain in each of those varied positions4 ,it is however to be understood that the different cases all presuppose the same total moment of momentum4 ,among the different cases that can be imagined1 those will be of special interest in which the total quantity of energy in the system is a maximum or a minimum4 ,we must for this purpose suppose the system gradually to run through all conceivable changes1 with the earth and moon as near as possible1 and as far as possible1 and in all intermediate positions2 we must also attribute to the earth every variety in the velocity of its rotation which is compatible with the preservation of the moment of momentum4 ,beginning then with the earth's velocity of rotation at its lowest1 we may suppose it gradually and continually increased1 and1 as we have already seen1 the change in the energy of the system is to be expressed by multiplying the relative rotation into the change of the earth's angular velocity4 ,it follows from the principles we have already explained1 that the maximum or minimum energy is attained at the moment when the alteration is zero4 ,it therefore follows1 that the critical periods of the system will arise when the relative rotation is zero1 that is1 when the earth's rotation on its axis is performed with a velocity equal to that with which the moon revolves around the earth4 ,this is truly a singular condition of the earth-moon system2 the moon in such a case would revolve around the earth as if the two bodies were bound together by rigid bonds into what was practically a single solid body4 ,at the present moment no doubt to some extent this condition is realized1 because the moon always turns the same face to the earth 7a point on which we shall have something to say later on72 but in the original condition of the earth-moon system1 the earth would also constantly direct the same face to the moon1 a condition of things which is now very far from being realized4 ,it can be shown from the mathematical nature of the problem that there are four states of the earth-moon system in which this condition may be realized1 and which are also compatible with the conservation of the moment of momentum4 ,we can express what this condition implies in a somewhat more simple manner4 ,let us understand by the day the period of the earth's rotation on its axis1 whatever that may be1 and let us understand by the month the period of revolution of the moon around the earth1 whatever value it may have2 then the condition of maximum or minimum energy is attained when the day and the month have become equal to each other4 ,of the four occasions mathematically possible in which the day and the month can be equal1 there are only two which at present need engage our attention--one of these occurred near the beginning of the earth and moon's history1 the other remains to be approached in the immeasurably remote future4 ,the two remaining solutions are futile1 being what the mathematician would describe as imaginary4 ,there is a fundamental difference between the dynamical conditions in these critical epochs--in one of them the energy of the system has attained a maximum value1 and in the other the energy of the system is at a minimum value4 ,it is impossible to over-estimate the significance of these two states of the system4 ,i may recall the fundamental notion which every one has learned in mechanics1 as to the difference between stable and unstable equilibrium4 ,the conceivable possibility of making an egg stand on its end is a practical impossibility1 because nature does not like unstable equilibrium1 and a body departs therefrom on the least disturbance2 on the other hand1 stable equilibrium is the position in which nature tends to place everything4 ,a log of wood floating on a river might conceivably float in a vertical position with its end up out of the water1 but you never could succeed in so balancing it1 because no matter how carefully you adjusted the log1 it would almost instantly turn over when you left it free2 on the other hand1 when the log floats naturally on the water it assumes a horizontal position1 to which1 when momentarily displaced therefrom1 it will return if permitted to do so4 ,we have here an illustration of the contrast between stable and unstable equilibrium4 ,it will be found generally that a body is in equilibrium when its centre of gravity is at its highest point or at its lowest point2 there is1 however1 this important difference1 that when the centre of gravity is highest the equilibrium is unstable1 and when the centre of gravity is lowest the equilibrium is stable4 ,the potential energy of an egg poised on its end in unstable equilibrium is greater than when it lies on its side in stable equilibrium4 ,in fact1 energy must be expended to raise the egg from the horizontal position to the vertical2 while1 on the other hand1 work could conceivably be done by the egg when it passes from the vertical position to the horizontal4 ,speaking generally1 we may say that the stable position indicates low energy1 while a redundancy of that valuable agent is suggestive of instability4 ,we may apply similar principles to the consideration of the earth-moon system4 ,it is true that we have here a series of dynamical phenomena1 while the illustrations ,i have given of stable and unstable equilibrium relate only to statical problems2 but we can have dynamical stability and dynamical instability1 just as we can have stable and unstable equilibrium4 ,dynamical instability corresponds with the maximum of energy1 and dynamical stability to the minimum of energy4 ,at that primitive epoch1 when the energy of the earth-moon system was a maximum1 the condition was one of dynamical instability2 it was impossible that it should last4 ,but now mark how truly critical an occurrence this must have been in the history of the earth-moon system1 for have ,i not already explained that it is a necessary condition of the progress of tidal evolution that the energy of the system should be always declining8 ,but here our retrospect has conducted us back to a most eventful crisis1 in which the energy was a maximum1 and therefore cannot have been immediately preceded by a state in which the energy was greater still2 it is therefore impossible for the tidal evolution to have produced this state of things2 some other influence must have been in operation at this beginning of the earth-moon system4 ,thus there can be hardly a doubt that immediately preceding the critical epoch the moon originated from the earth in the way we have described4 ,note also that this condition1 being one of maximum energy1 was necessarily of dynamical instability1 it could not last2 the moon must adopt either of two courses--it must tumble back on the earth1 or it must start outwards4 ,now which course was the moon to adopt8 ,the case is analogous to that of an egg standing on its end--it will inevitably tumble one way or the other4 ,some infinitesimal cause will produce a tendency towards one side1 and to that side accordingly the egg will fall4 ,the earth-moon system was similarly in an unstable state1 an infinitesimal cause might conceivably decide the fate of the system4 ,we are necessarily in ignorance of what the determining cause might have been1 but the effect it produced is perfectly clear2 the moon did not again return to its mother earth1 but set out on that mighty career which is in progress to-day4 ,let it be noted that these critical epochs in the earth-moon history arise when and only when there is an absolute identity between the length of the month and the length of the day4 ,it may be proper therefore that ,i should provide a demonstration of the fact1 that the identity between these two periods must necessarily have occurred at a very early period in the evolution4 ,the law of ,kepler1 which asserts that the square of the periodic time is proportioned to the cube of the mean distance1 is in its ordinary application confined to a comparison between the revolutions of the several planets about the sun4 ,the periodic time of each planet is connected with its average distance by this law2 but there is another application of ,kepler's law which gives us information of the distance and the period of the moon in former stages of the earth-moon history4 ,although the actual path of the moon is of course an ellipse1 yet that ellipse is troubled1 as is well known1 by many disturbing forces1 and from this cause alone the actual path of the moon is far from being any of those simple curves with which we are so well acquainted4 ,even were the earth and the moon absolutely rigid particles1 perturbations would work all sorts of small changes in the pliant curve4 ,the phenomena of tidal evolution impart an additional element of complexity into the actual shape of the moon's path4 ,we now see that the ellipse is not merely subject to incessant deflections of a periodic nature1 it also undergoes a gradual contraction as we look back through time past2 but we may1 with all needful accuracy for our present purpose1 think of the path of the moon as a circle1 only we must attribute to that circle a continuous contraction of its radius the further and the further we look back4 ,the alteration in the radius will be even so slow1 that the moon will accomplish thousands of revolutions around the earth without any appreciable alteration in the average distance of the two bodies4 ,we can therefore think of the moon as revolving at every epoch in a circle of special radius1 and as accomplishing that revolution in a special time4 ,with this understanding we can now apply ,kepler's law to the several stages of the moon's past history4 ,the periodic time of each revolution1 and the mean distance at which that revolution was performed1 will be always connected together by the formula of ,kepler4 ,thus to take an instance in the very remote past4 ,let us suppose that the moon was at one hundred and twenty thousand miles instead of two hundred and forty thousand1 that is1 at half its present distance4 ,applying the law of ,kepler1 we see that the time of revolution must then have been only about ten days instead of the twenty-seven it is now4 ,still further1 let us suppose that the moon revolves in an orbit with one-tenth of the diameter it has at present1 then the cube of #aj being #ajjj1 and the square root of #ajjj being #ca4f1 it follows that the month must have been less than the thirty-first part of what it is at present1 that is1 it must have been considerably less than one of our present days4 ,thus you see the month is growing shorter and shorter the further we look back1 the day is also growing shorter and shorter2 but still ,i think we can show that there must have been a time when the month will have been at least as short as the day4 ,for let us take the most extreme case in which the moon shall have made the closest possible approximation to the earth4 ,two globes in contact will have a distance between their centres which is equal to the sum of their radii4 ,take the earth as having a radius of four thousand miles1 and the moon a radius of one thousand miles1 the two centres must at their shortest distance be five thousand miles apart1 that is1 the moon must then be at the forty-eighth part of its present distance from the earth4 ,now the cube of #dh is #aaj1eib1 and the square root of #aaj1eib is nearly #ccc1 therefore the length of the month will be one-three hundred and thirty-third part of the duration of the month at present2 in other words1 the moon must revolve around the earth in a period of somewhat about two hours4 ,it seems impossible that the day can ever have been as brief as this4 ,we have therefore proved that1 in the course of its contracting duration1 the moon must have overtaken the contracting day1 and that therefore there must have been a time when the moon was in the vicinity of the earth1 and having a day and month of equal period4 ,thus we have shown that the critical condition of dynamical instability must have occurred in the early period of the earth-moon history1 if the agents then in operation were those which we now know4 ,the further development of the subject must be postponed until the next lecture4 ,,lecture ,,ii4 ,starting from that fitting commencement of earth-moon history which the critical epoch affords1 we shall now describe the dynamical phenomena as the tidal evolution progressed4 ,the moon and the earth initially moved as a solid body1 each bending the same face towards the other2 but as the moon retreated1 and as tides began to be raised on the earth1 the length of the day began to increase1 as did also the length of the month4 ,we know1 however1 that the month increased more rapidly than the day1 so that a time was reached when the month was twice as long as the day2 and still both periods kept on increasing1 but not at equal rates1 for in progress of time the month grew so much more rapidly than the day1 that many days had to elapse while the moon accomplished a single revolution4 ,it is1 however1 only necessary for us to note those stages of the mighty progress which correspond to special events4 ,the first of such stages was attained when the month assumed its maximum ratio to the day4 ,at this time1 the month was about twenty-nine days1 and the epoch appears to have occurred at a comparatively recent date if we use such standards of time as tidal evolution requires2 though measured by historical standards1 the epoch is of incalculable antiquity4 ,i cannot impress upon you too often the enormous magnitude of the period of time which these phenomena have required for their evolution4 ,professor ,darwin's theory affords but little information on this point1 and the utmost we can do is to assign a minor limit to the period through which tidal evolution has been in progress4 ,it is certain that the birth of the moon must have occurred at least fifty million years ago1 but probably the true period is enormously greater than this4 ,if indeed we choose to add a cipher or two to the figure just printed1 ,i do not think there is anything which could tell us that we have over-estimated the mark4 ,therefore1 when ,i speak of the epoch in which the month possessed the greatest number of days as a recent one1 it must be understood that ,i am merely speaking of events in relation to the order of tidal evolution4 ,viewed from this standpoint1 we can show that the epoch is a recent one in the following manner4 ,at present the month consists of a little more than twenty-seven days1 but at this maximum period to which ,i have referred the month was about twenty-nine days2 from that it began to decline1 and the decline cannot have proceeded very far1 for even still there are only two days less in the month than at the time when the month had the greatest number of days4 ,it thus follows that the present epoch--the human epoch1 as we may call it--in the history of the earth has fallen at a time when the progress of tidal evolution is about half-way between the initial and the final stage4 ,i do not mean half-way in the sense of actual measurement of years2 indeed1 from this point it would seem that we cannot yet be nearly half-way1 for1 vast as are the periods of time that have elapsed since the moon first took its departure from the earth1 they fall far short of that awful period of time which will intervene between the present moment and the hour when the next critical state of earth-moon history shall have been attained4 ,in that state the day is destined once again to be equal to the month1 just as was the case in the initial stage4 ,the half-way stage will therefore in one sense be that in which the proportion of the month to the day culminates4 ,this is the stage which we have but lately passed2 and thus it is that at present we may be said to be almost half-way through the progress of tidal evolution4 ,my narrative of the earth-moon evolution must from this point forward cease to be retrospective4 ,having begun at that critical moment when the month and day were first equal1 we have traced the progress of events to the present hour4 ,what we have now to say is therefore a forecast of events yet to come4 ,so far as we can tell1 no agent is likely to interfere with the gradual evolution caused by the tides1 which dynamical principles have disclosed to us4 ,as the years roll on1 or perhaps1 ,i should rather say1 as thousands of years and millions of years roll on1 the day will continue to elongate1 or the earth to rotate more slowly on its axis4 ,but countless ages must elapse before another critical stage of the history shall be reached4 ,it is needless for me to ponder over the tedious process by which this interesting epoch is reached4 ,i shall rather sketch what the actual condition of our system will be when that moment shall have arrived4 ,the day will then have expanded from the present familiar twenty-four hours up to a day more than twice1 more than five1 even more than fifty times its present duration4 ,in round numbers1 we may say that this great day will occupy one thousand four hundred of our ordinary hours4 ,to realize the critical nature of the situation then arrived at1 we must follow the corresponding evolution through which the moon passes4 ,from its present distance of two hundred and forty thousand miles1 the moon will describe an ever-enlarging orbit2 and as it does so the duration of the month will also increase1 until at last a point will be reached when the month has become more than double its present length1 and has attained the particular value of one thousand four hundred hours4 ,we are specially to observe that this one-thousand four-hundred-hour month will be exactly reached when the day has also expanded to one thousand four hundred hours2 and the essence of this critical condition1 which may be regarded as a significant point of tidal evolution1 is that the day and the month have again become equal4 ,the day and the month were equal at the beginning1 the day and the month will be equal at the end4 ,yet how wide is the difference between the beginning and the end4 ,the day or the month at the end is some hundreds of times as long as the month or the day at the beginning4 ,i have already fully explained how1 in any stage of the evolutionary progress in which the day and the month became equal1 the energy of the system attained a maximum or a minimum value4 ,at the beginning the energy was a maximum2 at the end the energy will be a minimum4 ,the most important consequences follow from this consideration4 ,i have already shown that a condition of maximum energy corresponded to dynamic instability4 ,thus we saw that the earth-moon history could not have commenced without the intervention of some influence other than tides at the beginning4 ,now let us learn what the similar doctrine has to tell us with regard to the end4 ,the condition then arrived at is one of dynamical stability2 for suppose that the system were to receive a slight alteration1 by which the moon went out a little further1 and thus described a larger orbit1 and so performed more than its share of the moment of spin4 ,then the earth would have to do a little less spinning1 because1 under all circumstances1 the total quantity of spin must be preserved unaltered4 ,but the energy being at a minimum1 such a small displacement must of course produce a state of things in which the energy would be increased4 ,or if we conceived the moon to come in towards the earth1 the moon would then contribute less to the total moment of momentum4 ,it would therefore be incumbent on the earth to do more2 and accordingly the velocity of the earth's rotation would be augmented4 ,but this arrangement also could only be produced by the addition of some fresh energy to the system1 because the position from which the system is supposed to have been disturbed is one of minimum energy4 ,no disturbance of the system from this final position is therefore conceivable1 unless some energy can be communicated to it4 ,but this will demonstrate the utter incompetency of the tides to shift the system by a hair's breadth from this position2 for it is of the essence of the tides to waste energy by friction4 ,and the transformations of the system which the tides have caused are invariably characterized by a decline of energy1 the movements being otherwise arranged so that the total moment of momentum shall be preserved intact4 ,note1 how far we were justified in speaking of this condition as a final one4 ,it is final so far as the lunar tides are concerned2 and were the system to be screened from all outer interference1 this accommodation between the earth and the moon would be eternal4 ,there is indeed another way of demonstrating that a condition of the system in which the day has assumed equality with the month must necessarily be one of dynamical equilibrium4 ,we have shown that the energy which the tides demand is derived not from the mere fact that there are high tides and low tides1 but from the circumstance that these tides do rise and fall2 that in falling and rising they do produce currents2 and it is these currents which generate the friction by which the earth's velocity is slowly abated1 its energy wasted1 and no doubt ultimately dissipated as heat4 ,if therefore we can make the ebbing and the flowing of the tides to cease1 then our argument will disappear4 ,thus suppose1 for the sake of illustration1 that at a moment when the tides happened to be at high water in the ,thames1 such a change took place in the behaviour of the moon that the water always remained full in the ,thames1 and at every other spot on the earth remained fixed at the exact height which it possessed at this particular moment4 ,there would be no more tidal friction1 and therefore the system would cease to course through that series of changes which the existence of tidal friction necessitates4 ,but if the tide is always to be full in the ,thames1 then the moon must be always in the same position with respect to the meridian1 that is1 the moon must always be fixed in the heavens over ,london4 ,in fact1 the moon must then revolve around the earth just as fast as ,london does--the month must have the same length as the day4 ,the earth must then show the same face constantly to the moon1 just as the moon always does show the same face towards the earth2 the two globes will in fact revolve as if they were connected with invisible bonds1 which united them into a single rigid body4 ,we need therefore feel no surprise at the cessation of the progress of tidal evolution when the month and the day are equal1 for then the movement of moon-raised tides has ceased4 ,no doubt the same may be said of the state at the beginning of the history1 when the day and the month had the brief and equal duration of a few hours4 ,while the equality of the two periods lasted there could be no tides1 and therefore no progress in the direction of tidal evolution4 ,there is1 however1 the profound difference of stability and instability between the two cases2 the most insignificant disturbance of the system at the initial stage was sufficient to precipitate the revolving moon from its condition of dynamical equilibrium1 and to start the course of tidal evolution in full vigour4 ,if1 however1 any trifling derangement should take place in the last condition of the system1 so that the month and the day departed slightly from equality1 there would instantly be an ebbing and a flowing of the tides2 and the friction generated by these tides would operate to restore the equality because this condition is one of dynamical stability4 ,it will thus be seen with what justice we can look forward to the day and month each of fourteen hundred hours as a finale to the progress of the luni-tidal evolution4 ,throughout the whole of this marvellous series of changes it is always necessary to remember the one constant and invariable element--the moment of momentum of the system which tides cannot alter4 ,whatever else the friction can have done1 however fearful may have been the loss of energy by the system1 the moment of momentum which the system had at the beginning it preserves unto the end4 ,this it is which chiefly gives us the numerical data on which we have to rely for the quantitative features of tidal evolution4 ,we have made so many demands in the course of these lectures on the capacity of tidal friction to accomplish startling phenomena in the evolution of the earth-moon system1 that it is well for us to seek for any evidence that may otherwise be obtainable as to the capacity of tides for the accomplishment of gigantic operations4 ,i do not say that there is any doubt which requires to be dispelled by such evidence1 for as to the general outlines of the doctrine of tidal evolution which has been here sketched out there can be no reasonable ground for mistrust2 but nevertheless it is always desirable to widen our comprehension of any natural phenomena by observing collateral facts4 ,now there is one branch of tidal action to which ,i have as yet only in the most incidental way referred4 ,we have been speaking of the tides in the earth which are made to ebb and flow by the action of the moon2 we have now to consider the tides in the moon1 which are there excited by the action of the earth4 ,for between these two bodies there is a reciprocity of tidal-making energy--each of them is competent to raise tides in the other4 ,as the moon is so small in comparison with the earth1 and as the tides on the moon are of but little significance in the progress of tidal evolution1 it has been permissible for us to omit them from our former discussion4 ,but it is these tides on the moon which will afford us a striking illustration of the competency of tides for stupendous tasks4 ,the moon presents a monument to show what tides are able to accomplish4 ,illustration3 ,fig4 #c4--,the ,moon4 ,i must first1 however1 explain a difficulty which is almost sure to suggest itself when we speak of tides on the moon4 ,i shall be told that the moon contains no water on its surface1 and how then1 it will be said1 can tides ebb and flow where there is no sea to be disturbed8 ,there are two answers to this difficulty2 it is no doubt true that the moon seems at present entirely devoid of water in so far as its surface is exposed to us1 but it is by no means certain that the moon was always in this destitute condition4 ,there are very large features marked on its map as seas2 these regions are of a darker hue than the rest of the moon's surface1 they are large objects often many hundreds of miles in diameter1 and they form1 in fact1 those dark patches on the brilliant surface which are conspicuous to the unaided eye1 and are represented in ,fig4 #c4 ,viewed in a telescope these so-called seas1 while clearly possessing no water at the present time1 are yet widely different from the general aspect of the moon's surface4 ,it has often been supposed that great oceans once filled these basins1 and a plausible explanation has even been offered as to how the waters they once contained could have vanished4 ,it has been thought that as the mineral substances deep in the interior of our satellite assumed the crystalline form during the progress of cooling1 the demand for water of crystallization required for incorporation with the minerals was so great that the oceans of the moon became entirely absorbed4 ,it is1 however1 unnecessary for our present argument that this theory should be correct4 ,even if there never was a drop of water found on our satellite1 the tides in its molten materials would be quite sufficient for our purpose2 anything that tides could accomplish would be done more speedily by vast tides of flowing lava than by merely oceanic tides4 ,there can be no doubt that tides raised on the moon by the earth would be greater than the tides raised on the earth by the moon4 ,the question is1 however1 not a very simple one1 for it depends on the masses of both bodies as well as on their relative dimensions4 ,in so far as the masses are concerned1 the earth being more than eighty times as heavy as the moon1 the tides would on this account be vastly larger on the moon than on the earth4 ,on the other hand1 the moon's diameter being much less than that of the earth1 the efficiency of a tide-producing body in its action on the moon would be less than that of the same body at the same distance in its action on the earth2 but the diminution of the tides from this cause would be not so great as their increase from the former cause1 and therefore the net result would be to exhibit much greater tides on the moon than on the earth4 ,suppose that the moon had been originally endowed with a rapid movement of rotation around its axis1 the effect of the tides on that rotation would tend to check its velocity just in the same way as the tides on the earth have effected a continual elongation of the day4 ,only as the tides on the moon were so enormously great1 their capacity to check the moon's speed would have corresponding efficacy4 ,in addition to this1 the mass of the moon being so small1 it could only offer feeble resistance to the unceasing action of the tide1 and therefore our satellite must succumb to whatever the tides desired ages before our earth would have been affected to a like extent4 ,it must be noticed that the influence of the tidal friction is not directed to the total annihilation of the rotation of the two bodies affected by it2 the velocity is only checked down until it attains such a point that the speed in which each body rotates upon its axis has become equal to that in which it revolves around the tide-producer4 ,the practical effect of such an adjustment is to make the tide-agitated body turn a constant face towards its tormentor4 ,i may here note a point about which people sometimes find a little difficulty4 ,the moon constantly turns the same face towards the earth1 and therefore people are sometimes apt to think that the moon performs no rotation whatever around its own axis4 ,but this is indeed not the case4 ,the true inference to be drawn from the constant face of the moon is1 that the velocity of rotation about its own axis is equal to that of its rotation around the earth2 in fact1 the moon revolves around the earth in twenty-seven days1 and its rotation about its axis is performed in twenty-seven days also4 ,you may illustrate the movement of the moon around the earth by walking around a table in a room1 keeping all the time your face turned towards the table2 in such a case as this you not only perform a motion of revolution1 but you also perform a rotation in an equal period4 ,the proof that you do rotate is to be found in the fact that during the movement your face is being directed successively to all the points of the compass4 ,there is no more singular fact in the solar system than the constancy of the moon's face to the earth4 ,the periods of rotation and revolution are both alike2 if one of these periods exceeded the other by an amount so small as the hundredth part of a second1 the moon would in the lapse of ages permit us to see that other side which is now so jealously concealed4 ,the marvellous coincidence between these two periods would be absolutely inexplicable1 unless we were able to assign it to some physical cause4 ,it must be remembered that in this matter the moon occupies a unique position among the heavenly host4 ,the sun revolves around on its axis in a period of twenty-five or twenty-six days--thus we see one side of the sun as frequently as we see the other4 ,the side of the sun which is turned towards us to-day is almost entirely different from that we saw a fortnight ago4 ,nor is the period of the sun's rotation to be identified with any other remarkable period in our system4 ,if it were equal to the length of the year1 for instance1 or if it were equal to the period of any of the other planets1 then it could hardly be contended that the phenomenon as presented by the moon was unique2 but the sun's period is not simply related1 or indeed related at all1 to any of the other periodic times in the system4 ,nor do we find anything like the moon's constancy of face in the behaviour of the other planets4 ,jupiter turns now one face to us and then another4 ,nor is his rotation related to the sun or related to any other body1 as our moon's motion is related to us4 ,it has indeed been thought that in the movements of the satellites of ,jupiter a somewhat similar phenomenon may be observed to that in the motion of our own satellite4 ,if this be so1 the causes whereby this phenomenon is produced are doubtless identical in the two cases4 ,so remarkable a coincidence as that which the moon's motion shows could not reasonably be explained as a mere fortuitous circumstance2 nor need we hesitate to admit that a physical explanation is required when we find a most satisfactory one ready for our acceptance1 as was originally pointed out by ,helmholtz4 ,there can be no doubt whatever that the constancy of the moon's face is the work of ancient tides1 which have long since ceased to act4 ,we have shown that if the moon's rotation had once been too rapid to permit of the same face being always directed towards us1 the tides would operate as a check by which the velocity of that rotation would be abated4 ,on the other hand1 if the moon rotated so slowly that its other face would be exposed to us in the course of the revolution1 the tides would then be dragged violently over its surface in the direction of its rotation2 their tendency would thus be to accelerate the speed until the angular velocity of rotation was equal to that of revolution4 ,thus the tides would act as a controlling agent of the utmost stringency to hurry the moon round when it was not turning fast enough1 and to arrest the motion when going too fast4 ,peace there would be none for the moon until it yielded absolute compliance to the tyranny of the tides1 and adjusted its period of rotation with exact identity to its period of revolution4 ,doubtless this adjustment was made countless ages ago1 and since that period the tides have acted so as to preserve the adjustment1 as long as any part of the moon was in a state sufficiently soft or fluid to respond to tidal impression4 ,the present state of the moon is a monument to which we may confidently appeal in support of our contention as to the great power of the tides during the ages which have passed2 it will serve as an illustration of the future which is reserved for our earth in ages yet to come1 when our globe shall have also succumbed to tidal influence4 ,it is owing to the smallness of the moon relatively to the earth that the tidal process has reached a much more advanced stage in the moon than it has on the earth2 but the moon is incessant in its efforts to bring the earth into the same condition which it has itself been forced to assume4 ,thus again we look forward to an epoch in the inconceivably remote future when tidal thraldom shall be supreme1 and when the earth shall turn the same face to the moon1 as the moon now turns the same face to the earth4 ,in the critical state of things thus looming in the dim future1 the earth and the moon will continue to perform this adjusted revolution in a period of about fourteen hundred hours1 the two bodies being held1 as it were1 by invisible bands4 ,such an arrangement might be eternal if there were no intrusion of tidal influence from any other body2 but of course in our system as we actually find it the sun produces tides as well as the moon2 and the solar tides being at present much less than those originated by the moon1 we have neglected them in the general outlines of the theory4 ,the solar tides1 however1 must necessarily have an increasing significance4 ,i do not mean that they will intrinsically increase1 for there seems no reason to apprehend any growth in their actual amount2 it is their relative importance to the lunar tides that is the augmenting quantity4 ,as the final state is being approached1 and as the velocity of the earth's rotation is approximating to the angular velocity with which the moon revolves around it1 the ebbing and the flowing of the lunar tides must become of evanescent importance2 and this indeed for a double reason1 partly on account of the moon's greatly augmented distance1 and partly on account of the increasing length of the lunar day1 and the extremely tardy movements of ebb and flow that the lunar tides will then have4 ,thus the lunar tides1 so far as their dynamical importance is concerned1 will ultimately become zero1 while the solar tides retain all their pristine efficiency4 ,we have therefore to examine the dynamical effects of solar tides on the earth and moon in the critical stage to which the present course of things tends4 ,the earth will then rotate in a period of about fifty-seven of its present days2 and considering that the length of the day1 though so much greater than our present day1 is still much less than the year1 it follows that the solar tides must still continue so as to bring the earth's velocity of rotation to a point even lower than it has yet attained4 ,in fact1 if we could venture to project our glance sufficiently far into the future1 it would seem that the earth must ultimately have its velocity checked by the sun-raised tides1 until the day itself had become equal to the year4 ,the dynamical considerations become1 however1 too complex for us to follow them1 so that ,i shall be content with merely pointing out that the influence of the solar tides will prevent the earth and moon from eternally preserving the relations of bending the same face towards each other2 the earth's motion will1 in fact1 be so far checked1 that the day will become longer than the month4 ,thus the doctrine of tidal evolution has conducted us to a prospect of a condition of things which will some time be reached1 when the moon will have receded to a distance in which the month shall have become about fifty-seven days1 and when the earth around which this moon revolves shall actually require a still longer period to accomplish its rotation on its axis4 ,here is an odd condition for a planet with its satellite2 indeed1 until a dozen years ago it would have been pronounced inconceivable that a moon should whirl round a planet so quickly that its journey was accomplished in less than one of the planet's own days4 ,arguments might be found to show that this was impossible1 or at least unprecedented4 ,there is our own moon1 which now takes twenty-seven days to go round the earth2 there is ,jupiter1 with four moons1 and the nearest of these to the primary goes round in forty-two and a half hours4 ,no doubt this is a very rapid motion2 but all those matters are much more lively with ,jupiter than they are here4 ,the giant planet himself does not need ten hours for a single rotation1 so that you see his nearest moon still takes between two and three ,jovian days to accomplish a single revolution4 ,the example of ,saturn might have been cited to show that the quickest revolution that any satellite could perform must still require at least twice as long as the day in which the planet performed its rotation4 ,nor could the rotation of the planets around the sun afford a case which could be cited4 ,for even ,mercury1 the nearest of all the planets to the sun of which the existence is certainly known1 and therefore the most rapid in its revolution1 requires eighty-eight days to get round once2 and in the mean time the sun has had time to accomplish between three and four rotations4 ,indeed1 the analogies would seem to have shown so great an improbability in the conclusion towards which tidal evolution points1 that they would have contributed a serious obstacle to the general acceptance of that theory4 ,but in #ahgg an event took place so interesting in astronomical history1 that we have to look back to the memorable discovery of ,uranus in #agha before we can find a parallel to it in importance4 ,mars had always been looked upon as one of the moonless planets1 though grounds were not wanting for the surmise that probably moons to ,mars really existed4 ,it was under the influence of this belief that an attempt was made by ,professor ,asaph ,hall at ,washington to make a determined search1 and see if ,mars might not be attended by satellites large enough to be discoverable4 ,the circumstances under which this memorable inquiry was undertaken were eminently favourable for its success4 ,the orbit of ,mars is one which possesses an exceptionally high eccentricity2 it consequently happens that the oppositions during which the planet is to be observed vary very greatly in the facilities they afford for a search like that contemplated by ,professor ,hall4 ,it is obviously advantageous that the planet should be situated as near as possible to the earth1 and in the opposition in #ahgg the distance was almost at the lowest point it is capable of attaining2 but this was not the only point in which ,professor ,hall was favoured2 he had the use of a telescope of magnificent proportions and of consummate optical perfection4 ,his observatory was also placed in ,washington1 so that he had the advantage of a pure sky and of a much lower latitude than any observatory in ,great ,britain is placed at4 ,but the most conspicuous advantage of all was the practised skill of the astronomer himself1 without which all these other advantages would have been but of little avail4 ,great success rewarded his well-designed efforts2 not alone was one satellite discovered which revolved around the planet in a period conformable with that of other similar cases1 but a second little satellite was found1 which accomplished its revolution in a wholly unexpected and unprecedented manner4 ,the day of ,mars himself1 that is1 the period in which he can accomplish a rotation around his axis1 very closely approximates to our own day1 being in fact half an hour longer4 ,this little satellite1 the inner and more rapid of the pair1 requires for a single revolution a period of only seven hours thirty-nine minutes1 that is to say1 the little body scampers more than three times round its primary before the primary itself has finished one of its leisurely rotations4 ,here was indeed a striking fact1 a unique fact in our system1 which riveted the attention of astronomers on this most beautiful discovery4 ,you will now see the bearing which the movement of the inner satellite of ,mars has on the doctrine of tidal evolution4 ,as a legitimate consequence of that doctrine1 we came to the conclusion that our earth-moon system must ultimately attain a condition in which the day is longer than the month4 ,but this conclusion stood unsupported by any analogous facts in the more anciently-known truths of astronomy4 ,the movement of the satellite of ,mars1 however1 affords the precise illustration we want2 and this fact1 ,i think1 adds an additional significance to the interest and the beauty of ,professor ,hall's discovery4 ,it is of particular interest to investigate the possible connection which the phenomena of tidal evolution may have had in connection with the geological phenomena of the earth4 ,we have already pointed out the greater closeness of the moon to us in times past4 ,the tides raised by the moon on the earth must therefore have been greater in past ages than they are now1 for of course the nearer the moon the bigger the tide4 ,as soon as the earth and the moon had separated to a considerable distance we may say that the height of the tide will vary inversely as the cube of the moon's distance2 it will therefore happen1 that when the moon was at half its present distance from us1 his tide-producing capacity was not alone twice as much or four times as much1 but even eight times as much as it is at present2 and a much greater rate of tidal rise and fall indicates1 of course1 a preponderance in every other manifestation of tidal activity4 ,the tidal currents1 for instance1 must have been much greater in volume and in speed2 even now there are places in which the tidal currents flow at four or more miles per hour4 ,we can imagine1 therefore1 the vehemence of the tidal currents which must have flowed in those days when the moon was a much smaller distance from us4 ,it is interesting to view these considerations in their possible bearings on geological phenomena4 ,it is true that we have here many elements of uncertainty1 but there is1 however1 a certain general outline of facts which may be laid down1 and which appears to be instructive1 with reference to the past history of our earth4 ,i have all through these lectures indicated a mighty system of chronology for the earth-moon system4 ,it is true that we cannot give our chronology any accurate expression in years4 ,the various stages of this history are to be represented by the successive distances between the earth and the moon4 ,each successive epoch1 for instance1 may be marked by the number of thousands of miles which separate the moon from the earth4 ,but we have another system of chronology derived from a wholly different system of ideas2 it too relates to periods of vast duration1 and1 like our great tidal periods1 extends to times anterior to human history1 or even to the duration of human life on this globe4 ,the facts of geology open up to us a majestic chronology1 the epochs of which are familiar to us by the succession of strata forming the crust of the earth1 and by the succession of living beings whose remains these strata have preserved4 ,from the present or recent age our retrospect over geological chronology leads us to look through a vista embracing periods of time overwhelming in their duration1 until at last our view becomes lost1 and our imagination is baffled in the effort to comprehend the formation of those vast stratified rocks1 a dozen miles or more in thickness1 which seem to lie at the very base of the stratified system on the earth1 and in which it would appear that the dawnings of life on this globe may be almost discerned4 ,we have thus the two systems of chronology to compare--one1 the astronomical chronology measured by the successive stages in the gradual retreat of the moon2 the other1 the geological chronology measured by the successive strata constituting the earth's crust4 ,never was a more noble problem proposed in the physical history of our earth than that which is implied in the attempt to correlate these two systems of chronology4 ,what we would especially desire to know is the moon's distance which corresponds to each of the successive strata on the earth4 ,how far off1 for instance1 was that moon which looked down on the coal forests in the time of their greatest luxuriance8 or what was the apparent size of the full moon at which the ichthyosaurus could have peeped when he turned that wonderful eye of his to the sky on a fine evening8 ,but interesting as this great problem is1 it lies1 alas6 outside the possibility of exact solution4 ,indeed we shall not make any attempt which must necessarily be futile to correlate these chronologies2 all we can do is to state the one fact which is absolutely undeniable in the matter4 ,let us fix our attention on that specially interesting epoch at the dawn of geological time1 when those mighty ,laurentian rocks were deposited of which the thickness is so astounding1 and let us consider what the distance of the moon must have been at this initial epoch of the earth's history4 ,all we know for certain is1 that the moon must have been nearer1 but what proportion that distance bore to the present distance is necessarily quite uncertain4 ,some years ago ,i delivered a lecture at ,birmingham1 entitled ,a ,glimpse through the ,corridors of ,time1 and in that lecture ,i threw out the suggestion that the moon at this primeval epoch may have only been at a small fraction of its present distance from us1 and that consequently terrific tides may in these days have ravaged the coast4 ,there was a good deal of discussion on the subject1 and while it was universally admitted that the tides must have been larger in palozoic times than they are at present1 yet there was a considerable body of opinion to the effect that the tides even then may have been only about twice1 or possibly not so much1 greater than those tides we have at the present4 ,what the actual fact may be we have no way of knowing2 but it is interesting to note that even the smallest accession to the tides would be a valuable factor in the performance of geological work4 ,for let me recall to your minds a few of the fundamental phenomena of geology4 ,those stratified rocks with which we are now concerned have been chiefly manufactured by deposition of sediment in the ocean4 ,rivers1 swollen1 it may be1 by floods1 and turbid with a quantity of material held in suspension1 discharge their waters into the sea4 ,granting time and quiet1 this sediment falls to the bottom2 successive additions are made to its thickness during centuries and thousands of years1 and thus beds are formed which in the course of ages consolidate into actual rock4 ,in the formation of such beds the tides will play a part4 ,into the estuaries at the mouths of rivers the tides hurry in and hurry out1 and especially during spring tides there are currents which flow with tremendous power2 then too1 as the waves batter against the coast they gradually wear away and crumble down the mightiest cliffs1 and waft the sand and mud thus produced to augment that which has been brought down by the rivers4 ,in this operation also the tides play a part of conspicuous importance1 and where the ebb and flow is greatest it is obvious that an additional impetus will be given to the manufacture of stratified rocks4 ,in fact1 we may regard the waters of the globe as a mighty mill1 incessantly occupied in grinding up materials for future strata4 ,the main operating power of this mill is of course derived from the sun1 for it is the sun which brings up the rains to nourish the rivers1 it is the sun which raises the wind which lashes the waves against the shore4 ,but there is an auxiliary power to keep the mill in motion1 and that auxiliary power is afforded by the tides4 ,if then we find that by any cause the efficiency of the tides is increased we shall find that the mill for the manufacture of strata obtains a corresponding accession to its capacity4 ,assuming the estimate of ,professor ,darwin1 that the tide may have had twice as great a vertical range of ebb and flow within geological times as it has at present1 we find a considerable addition to the efficiency of the ocean in the manufacture of the ancient stratified rocks4 ,it must be remembered that the extent of the area through which the tides will submerge and lay bare the country1 will often be increased more than twofold by a twofold increase of height4 ,a little illustration may show what ,i mean4 ,suppose a cone to be filled with water up to a certain height1 and that the quantity of water in it be measured2 now let the cone be filled until the water is at double the depth2 then the surfaces of the water in the two cases will be in the ratio of the circles1 one of which has double the diameter of the other4 ,the areas of the two surfaces are thus as four to one2 the volumes of the waters in the two cases will be in the proportion of two similar solids1 the ratios of their dimensions being as two to one4 ,of course this means that the water in the one case would be eight times as much as in the other4 ,this particular illustration will not often apply exactly to tidal phenomena1 but ,i may mention one place that ,i happen to know of1 in the vicinity of ,dublin1 in which the effect of the rise and fall of the tide would be somewhat of this description4 ,at ,malahide there is a wide shallow estuary cut off from the sea by a railway embankment1 and there is a viaduct in the embankment through which a great tidal current flows in and out alternately4 ,at low tide there is but little water in this estuary1 but at high tide it extends for miles inland4 ,we may regard this inlet with sufficient approximation to the truth as half of a cone with a very large angle1 the railway embankment of course forming the diameter2 hence it follows that if the tide was to be raised to double its height1 so large an area of additional land would be submerged1 and so vast an increase of water would be necessary for the purpose1 that the flow under the railway bridge would have to be much more considerable than it is at present4 ,in some degree the same phenomena will be repeated elsewhere around the coast4 ,simply multiplying the height of the tide by two would often mean that the border of land between high and low water would be increased more than twofold1 and that the volume of water alternately poured on the land and drawn off it would be increased in a still larger proportion4 ,the velocity of all tidal currents would also be greater than at present1 and as the power of a current of water for transporting solid material held in suspension increases rapidly with the velocity1 so we may infer that the efficiency of tidal currents as a vehicle for the transport of comminuted rocks would be greatly increased4 ,it is thus obvious that tides with a rise and fall double in vertical height of those which we know at present would add a large increase to their efficiency as geological agents4 ,indeed1 even were the tides only half or one-third greater than those we know now1 we might reasonably expect that the manufacture of stratified rocks must have proceeded more rapidly than at present4 ,the question then will assume this form4 ,we know that the tides must have been greater in ,cambrian or ,laurentian days than they are at present2 so that they were available as a means of assisting other agents in the stupendous operations of strata manufacture which were then conducted4 ,this certainly helps us to understand how these tremendous beds of strata1 a dozen miles or more in solid thickness1 were deposited4 ,it seems imperative that for the accomplishment of a task so mighty1 some agents more potent than those with which we are familiar should be required4 ,the doctrine of tidal evolution has shown us what those agents were4 ,it only leaves us uninformed as to the degree in which their mighty capabilities were drawn upon4 ,it is the property of science as it grows to find its branches more and more interwoven1 and this seems especially true of the two greatest of all natural sciences--geology and astronomy4 ,with the beginnings of our earth as a globe in the shape in which we find it both these sciences are directly concerned4 ,i have here touched upon another branch in which they illustrate and confirm each other4 ,as the theory of tidal evolution has shed such a flood of light into the previously dark history of our earth-moon system1 it becomes of interest to see whether the tidal phenomena may not have a wider scope2 whether they may not1 for instance1 have determined the formation of the planets by birth from the sun1 just as the moon seems to have originated by birth from the earth4 ,our first presumption1 that the cases are analogous1 is not however justified when the facts are carefully inquired into4 ,a principle which ,i have not hitherto discussed here assumes prominence1 and therefore we shall devote our attention to it for a few minutes4 ,let us understand what we mean by the solar system4 ,there is first the sun at the centre1 which preponderates over all the other bodies so enormously1 as shown in ,fig4 #d1 in which the earth and the sun are placed side by side for comparison4 ,there is then the retinue of planets1 among the smaller of which our earth takes its place1 a view of the comparative sizes of the planets being shown in ,fig4 #e4 ,illustration3 ,fig4 #d4--;,comparative sizes of ,earth and ,sun4 ,not to embarrass ourselves with the perplexities of a problem so complicated as our solar system is in its entirety1 we shall for the sake of clear reasoning assume an ideal system1 consisting of a sun and a large planet--in fact1 such as our own system would be if we could withdraw from it all other bodies1 leaving the sun and ,jupiter only remaining4 ,we shall suppose1 of course1 that the sun is much larger than the planet1 in fact1 it will be convenient to keep in mind the relative masses of the sun and ,jupiter1 the weight of the planet being less than one-thousandth part of the sun4 ,we know1 of course1 that both of those bodies are rotating upon their axes1 and the one is revolving around the other2 and for simplicity we may further suppose that the axes of rotation are perpendicular to the plane of revolution4 ,in bodies so constituted tides will be manifested4 ,jupiter will raise tides in the sun1 the sun will raise tides in ,jupiter4 ,if the rotation of each body be performed in a less period than that of the revolution 7the case which alone concerns us71 then the tides will immediately operate in their habitual manner as a brake for the checking of rotation4 ,the tides raised by the sun on ,jupiter will tend therefore to lengthen ,jupiter's day2 the tides raised on the sun by ,jupiter will tend to augment the sun's period of rotation4 ,both ,jupiter and the sun will therefore lose some moment of momentum4 ,we cannot1 however1 repeat too often the dynamical truth that the total moment of momentum must remain constant1 therefore what is lost by the rotation must be made up in the revolution2 the orbit of ,jupiter around the sun must accordingly be swelling4 ,so far the reasoning appears similar to that which led to such startling consequences in regard to the moon4 ,illustration3 ,fig4 #e4--;,comparative sizes of ,planets4 ,but now for the fundamental difference between the two cases4 ,the moon1 it will be remembered1 always revolves with the same face towards the earth4 ,the tides have ceased to operate there1 and consequently the moon is not able to contribute any moment of momentum1 to be applied to the enlargement of its distance from the earth2 all the moment of momentum necessary for this purpose is of course drawn from the single supply in the rotation of the earth on its axis4 ,but in the case of the system consisting of the sun and ,jupiter the circumstances are quite different--,jupiter does not always bend the same face to the sun2 so far1 indeed1 is this from being true1 that ,jupiter is eminently remarkable for the rapidity of his rotation1 and for the incessantly varying aspect in which he would be seen from the sun4 ,jupiter has therefore a store of available moment of momentum1 as has also of course the sun4 ,thus in the sun and planet system we have in the rotations two available stores of moment of momentum on which the tides can make draughts for application to the enlargement of the revolution4 ,the proportions in which these two available sources can be drawn upon for contributions is not left arbitrary4 ,the laws of dynamics provide the shares in which each of the bodies is to contribute for the joint purpose of driving them further apart4 ,let us see if we cannot form an estimate by elementary considerations as to the division of the labour4 ,the tides raised on ,jupiter by the sun will be practically proportional to the sun's mass and to the radius of ,jupiter4 ,owing to the enormous size of the sun1 the efficiency of these tides and the moment of the friction-brake they produce will be far greater on the planet than will the converse operation of the planet be on the sun4 ,hence it follows that the efficiency of the tides in depriving ,jupiter of moment of momentum will be greatly superior to the efficiency of the tides in depriving the sun of moment of momentum4 ,without following the matter into any close numerical calculation1 we may assert that for every one part the sun contributes to the common object1 ,jupiter will contribute at least a thousand parts2 and this inequality appears all the more striking1 not to say unjust1 when it is remembered that the sun is more abundantly provided with moment of momentum than is ,jupiter--the sun has1 in fact1 about twenty thousand times as much4 ,the case may be illustrated by supposing that a rich man and a poor man combine together to achieve some common purpose to which both are to contribute4 ,the ethical notion that ,dives shall contribute largely1 according to his large means1 and ,lazarus according to his slender means1 is quite antagonistic to the scale which dynamics has imposed4 ,dynamics declares that the rich man need only give a penny to every pound that has to be extorted from the poor man4 ,now this is precisely the case with regard to the sun and ,jupiter1 and it involves a somewhat curious consequence4 ,as long as ,jupiter possesses available moment of momentum1 we may be certain that no large contribution of moment of momentum has been obtained from the sun4 ,for1 returning to our illustration1 if we find that ,lazarus still has something left in his pocket1 we are of course assured that ,dives cannot have expended much1 because1 as ,lazarus had but little to begin with1 and as ,dives only puts in a penny for every pound that ,lazarus spends1 it is obvious that no large amount can have been devoted to the common object4 ,hence it follows that whatever transfer of moment of momentum has taken place in the sun-,jupiter system has been almost entirely obtained at the expense of ,jupiter4 ,now in the solar system at present1 the orbital moment of momentum of ,jupiter is nearly fifty thousand times as great as his present store of rotational moment of momentum4 ,if1 therefore1 the departure of ,jupiter from the sun had been the consequence of tidal evolution1 it would follow that ,jupiter must once have contained many thousands of times the moment of momentum that he has at present4 ,this seems utterly incredible1 for even were ,jupiter dilated into an enormously large mass of vaporous matter1 spinning round with the utmost conceivable speed1 it is impossible that he should ever have possessed enough moment of momentum4 ,we are therefore forced to the conclusion that the tides alone do not provide sufficient explanation for the retreat of ,jupiter from the sun4 ,there is rather a subtle point in the considerations now brought forward1 on which it will be necessary for us to ponder4 ,in the illustration of ,dives and ,lazarus1 the contributions of ,lazarus of course ceased when his pockets were exhausted1 but those of ,dives will continue1 and in the lapse of time may attain any amount within the utmost limits of ,dives' resources4 ,the essential point to notice is1 that so long as ,lazarus retains anything in his pocket1 we know for certain that ,dives has not given much2 if ,lazarus1 however1 has his pocket absolutely empty1 and if we do not know how long they may have been in that condition1 we have no means of knowing how large a portion of wealth ,dives may not have actually expended4 ,the turning-point of the theory thus involves the fact that ,jupiter still retains available moment of momentum in his rotation2 and this was our sole method of proving that the sun1 which in this case was ,dives1 had never given much4 ,but our argument must have taken an entirely different line had it so happened that ,jupiter constantly turned the same face to the sun1 and that therefore his pockets were entirely empty in so far as available moment of momentum is concerned4 ,it would be apparently impossible for us to say to what extent the resources of the sun may not have been drawn upon2 we can1 however1 calculate whether in any case the sun could possibly have supplied enough moment of momentum to account for the recession of ,jupiter4 ,speaking in round numbers1 the revolutional moment of momentum of ,jupiter is about thirty times as great as the rotational moment of momentum at present possessed by the sun4 ,i do not know that there is anything impossible in the supposition that the sun might1 by an augmented volume and an augmented velocity of rotation1 contain many times the moment of momentum that it has at this moment4 ,it therefore follows that if it had happened that ,jupiter constantly bent the same face to the sun1 there would apparently be nothing impossible in the fact that ,jupiter had been born of the sun1 just as the moon was born of the earth4 ,these same considerations should also lead us to observe with still more special attention the development of the earth-moon system4 ,let us restate the matter of the earth and moon in the light which the argument with respect to ,jupiter has given us4 ,at present the rotational moment of momentum of the earth is about a fifth part of the revolutional moment of momentum of the moon4 ,owing to the fact that the moon keeps the same face to us1 she has now no available moment of momentum1 and all the moment of momentum required to account for her retreat has of late come from the rotation of the earth2 but suppose that the moon still had some liquid on its surface which could be agitated by tides1 suppose further that it did not always bend the same face towards us1 that it therefore had some available moment of momentum due to its rotation on which the tides could operate1 then see how the argument would have been altered4 ,the gradual increase of the moon's distance could be provided for by a transfer of moment of momentum from two sources1 due of course to the rotational velocities of the two bodies4 ,here again the moon and the earth will contribute according to that dynamical but very iniquitous principle which regulated the appropriations from the purses of ,dives and ,lazarus4 ,the moon must give not according to her abundance1 but in the inverse ratio thereof--because she has little she must give largely4 ,nor shall we make an erroneous estimate if we say that nine-tenths of the whole moment of momentum necessary for the enlargement of the orbit would have been exacted from the moon2 that means that the moon must once have had about five or six times as much moment of momentum as the earth possesses at this moment4 ,considering the small size of the moon1 this could only have arisen by terrific velocity of rotation1 which it is inconceivable that its materials could ever have possessed4 ,this presents the demonstration of tidal evolution in a fresh light4 ,if the moon now departed to any considerable extent from showing the constant face to the earth1 it would seem that its retreat could not have been caused by tides4 ,some other agent for producing the present configuration would be necessary1 just as we found that some other agent than the tides has been necessary in the case of ,jupiter4 ,but ,i must say a few words as to the attitude of this question with regard to the entire solar system4 ,this system consists of the sun presiding at the centre1 and of the planets and their satellites in revolution around their respective primaries1 and each also animated by a rotation on its axis4 ,i shall in so far depart from the actual configuration of the system as to transform it into an ideal system1 whereof the masses1 the dimensions1 and the velocities shall all be preserved2 but that the several planes of revolution shall be all flattened into one plane1 instead of being inclined at small angles as they are at present2 nor will it be unreasonable for us at the same time to bring into parallelism all the axes of rotation1 and to arrange that their common directions shall be perpendicular to the plane of their common orbits4 ,for the purpose of our present research this ideal system may pass for the real system4 ,in its original state1 whatever that state may have been1 a magnificent endowment was conferred upon the system4 ,perhaps ,i may1 without derogation from the dignity of my subject1 speak of the endowment as partly personal and partly entailed4 ,the system had of course different powers with regard to the disposal of the two portions2 the personal estate could be squandered4 ,it consisted entirely of what we call energy2 and considering how frequently we use the expression conservation of energy1 it may seem strange to say now that this portion of the endowment has been found capable of alienation1 nay1 further1 that our system has been squandering it persistently from the first moment until now4 ,although the doctrine of the conservation of energy is1 we have every reason to believe1 a fundamental law affecting the whole universe1 yet it would be wholly inaccurate to say that any particular system such as our solar system shall invariably preserve precisely the same quantity of energy without alteration4 ,the circumstance that heat is a form of energy indeed negatives this supposition4 ,for our system possesses energy of all the different kinds3 there is energy due to the motions both of rotation and of revolution2 there is energy due to the fact that the mutually attracting bodies of our system are separated by distances of enormous magnitude2 and there is also energy in the form of heat2 and the laws of heat permit that this form of energy shall be radiated off into space1 and thus disappear entirely1 in so far as our system is concerned4 ,on the other hand1 there may no doubt be some small amount of energy accruing to our system from the other systems in space1 which like ours are radiating forth energy4 ,any gain from this source1 however1 is necessarily so very small in comparison to the loss to which we have referred1 that it is quite impossible that the one should balance the other4 ,though it is undoubtedly true that the total quantity of energy in the universe is constant1 yet the share of that energy belonging to any particular system such as ours declines steadily from age to age4 ,i may indeed remark1 that the question as to what becomes of all the radiant energy which the millions of suns in the universe are daily discharging offers a problem apparently not easy to solve2 but we need not discuss the matter at present1 we are only going to trace out the vicissitudes of our own system2 and whatever other changes that system may exhibit1 the fact is certain that the total quantity of energy it contains is declining4 ,of the two endowments of energy and of moment of momentum originally conferred on our system the moment of momentum is the entailed estate4 ,no matter how the bodies may move1 no matter how their actions may interfere with one another1 no matter how this body is pulled one way and the other body that way1 the conservation of moment of momentum is not imperilled1 nor1 no matter what losses of heat may be experienced by radiation1 could the store of moment of momentum be affected4 ,the only conceivable way in which the quantity of moment of momentum in the solar system could be tampered with is by the interference of some external attracting body4 ,we know1 however1 that the stars are all situated at such enormous distances1 that the influences they can exert in the perturbation of the solar system are absolutely insensible2 they are beyond the reach of the most delicate astronomical measurements4 ,hence we see how the endowment of the system with moment of momentum has conferred upon that system a something which is absolutely inalienable1 even to the smallest portion4 ,before going any further it would be necessary for me to explain more fully than ,i have hitherto done the true nature of the method of estimating moment of momentum4 ,the moment of momentum consists of two parts3 there is first that due to the revolution of the bodies around the sun2 there is secondly the rotation of these bodies on their axes4 ,let us first think simply of a single planet revolving in a circular orbit around the sun4 ,the momentum of that planet at any moment may be regarded as the product of its mass and its velocity2 then the moment of momentum of the planet in the case mentioned is found by multiplying the momentum by the radius of the path pursued4 ,in a more general case1 where the planet does not revolve in a circle1 but pursued an elliptic path1 the moment of momentum is to be found by multiplying the planet's velocity and its mass into the perpendicular from the sun on the direction in which the planet is moving4 ,these rules provide the methods for estimating all the moments of momentum1 so far as the revolutions in our system are concerned4 ,for the rotations somewhat more elaborate processes are required4 ,let us think of a sphere rotating round a fixed axis4 ,every particle of that sphere will of course describe a circle around the axis1 and all these circles will lie in parallel planes4 ,we may for our present purpose regard each atom of the body as a little planet revolving in a circular orbit1 and therefore the moment of momentum of the entire sphere will be found by simply adding together the moments of momentum of all the different atoms of which the sphere is composed4 ,to perform this addition the use of an elaborate mathematical method is required4 ,i do not propose to enter into the matter any further1 except to say that the total moment of momentum is the product of two factors--one the angular velocity with which the sphere is turning round1 while the other involves the sphere's mass and dimensions4 ,to illustrate the principles of the computation we shall take one or two examples4 ,suppose that two circles be drawn1 one of which is double the diameter of the other4 ,let two planets be taken of equal mass1 and one of these be put to revolve in one circle1 and the other to revolve in the other circle1 in such a way that the periods of both revolutions shall be equal4 ,it is required to find the moments of momentum in the two cases4 ,in the larger of the two circles it is plain that the planet must be moving twice as rapidly as in the smaller1 therefore its momentum is twice as great2 and as the radius is also double1 it follows that the moment of momentum in the large orbit will be four times that in the small orbit4 ,we thus see that the moment of momentum increases in the proportion of the squares of the radii4 ,if1 however1 the two planets were revolving about the same sun1 one of these orbits being double the other1 the periodic times could not be equal1 for ,kepler's law tells us that the square of the periodic time is proportional to the cube of the mean distance4 ,suppose1 then1 that the distance of the first planet is #a1 and that of the second planet is #b1 the cubes of those numbers are #a and #h1 and therefore the periodic times of the two bodies will be as #a to the square root of #h4 ,we can thus see that the velocity of the outer body must be less than that of the inner one1 for while the length of the path is only double as large1 the time taken to describe that path is the square root of eight times as great2 in fact1 the velocity of the outer body will be only the square root of twice that of the inner one4 ,as1 however1 its distance from the sun is twice as great1 it follows that the moment of momentum of the outer body will be the square root of twice that of the inner body4 ,we may state this result a little more generally as follows-- ,in comparing the moments of momentum of the several planets which revolve around the sun1 that of each planet is proportional to the product of its mass with the square root of its distance from the sun4 ,let us now compare two spheres together1 the diameter of one sphere being double that of the other1 while the times of rotation of the two are identical4 ,and let us now compare together the moments of momentum in these two cases4 ,it can be shown by reasoning1 into which ,i need not now enter1 that the moment of momentum of the large sphere will be thirty-two times that of the small one4 ,in general we may state that if a sphere of homogeneous material be rotating about an axis1 its moment of momentum is to be expressed by the product of its angular velocity by the fifth power of its radius4 ,we can now take stock1 as it were1 of the constituents of moments of momentum in our system4 ,we may omit the satellites for the present1 while such unsubstantial bodies as comets and such small bodies as meteors need not concern us4 ,the present investment of the moment of momentum of our system is to be found by multiplying the mass of each planet by the square root of its distance from the sun2 these products for all the several planets form the total revolutional moment of momentum4 ,the remainder of the investment is in rotational moment of momentum1 the collective amount of which is to be estimated by multiplying the angular velocity of each planet into its density1 and the fifth power of its radius if the planet be regarded as homogeneous1 or into such other power as may be necessary when the planet is not homogeneous4 ,indeed1 as the denser parts of the planet necessarily lie in its interior1 and have therefore neither the velocity nor the radius of the more superficial portions1 it seems necessary to admit that the moments of momentum of the planets will be proportional to some lower power of the radius than the fifth4 ,the total moment of momentum of the planets by rotation1 when multiplied by a constant factor1 and added to the revolutional moment of momentum1 will remain absolutely constant4 ,it may be interesting to note the present disposition of this vast inheritance among the different bodies of our system4 ,the biggest item of all is the moment of momentum of ,jupiter1 due to its revolution around the sun2 in fact1 in this single investment nearly sixty per cent4 of the total moment of momentum of the solar system is found4 ,the next heaviest item is the moment of momentum of ,saturn's revolution1 which is twenty-four per cent4 ,then come the similar contributions of ,uranus and ,neptune1 which are six and eight per cent4 respectively4 ,only one more item is worth mentioning1 as far as magnitude is concerned1 and that is the nearly two per cent4 that the sun contains in virtue of its rotation4 ,in fact1 all the other moments of momentum are comparatively insignificant in this method of viewing the subject4 ,jupiter from his rotation has not the fifty thousandth part of his revolutional moment of momentum1 while the earth's rotational share is not one ten thousandth part of that of ,jupiter1 and therefore is without importance in the general aspect of the system4 ,the revolution of the earth contributes about one eight hundredth part of that of ,jupiter4 ,these facts as here stated will suffice for us to make a forecast of the utmost the tides can effect in the future transformation of our system4 ,we have already explained that the general tendency of tidal friction is to augment revolutional moment of momentum at the expense of rotational4 ,the total1 however1 of the rotational moment of momentum of the system barely reaches two per cent4 of the whole amount2 this is of course almost entirely contributed by the sun1 for all the planets together have not a thousandth part of the sun's rotational moment4 ,the utmost therefore that tidal evolution can effect in the system is to distribute the two per cent4 in augmenting the revolutionary moment of momentum4 ,it does not seem that this can produce much appreciable derangement in the configuration of the system4 ,no doubt if it were all applied to one of the smaller planets it would produce very considerable effect4 ,our earth1 for instance1 would have to be driven out to a distance many hundreds of times further than it is at present were the sun's disposable moment of momentum ultimately to be transferred to the earth alone4 ,on the other hand1 ,jupiter could absorb the whole of the sun's share by quite an insignificant enlargement of its present path4 ,it does not seem likely that the distribution that must ultimately take place can much affect the present configuration of the system4 ,we thus see that the tides do not appear to have exercised anything like the same influence in the affairs of our solar system generally which they have done in that very small part of the solar system which consists of the earth and moon4 ,this is1 as ,i have endeavoured to show in these lectures1 the scene of supremely interesting tidal phenomena2 but how small it is in comparison with the whole magnitude of our system may be inferred from the following illustration4 ,i represent the whole moment of momentum of our system by #a1jjj1jjj1jjj1 the bulk of which is composed of the revolutional moments of momentum of the great planets1 and the rotational moment of momentum of the sun4 ,on this scale the rotational share which has fallen to our earth and moon does not even rise to the dignity of a single pound1 it can only be represented by the very modest figure of #ais4 #e;d4 ,this is divided into two parts--the earth by its rotation accounts for #cs4 #d;d41 leaving #afs4 #a;d4 as the equivalent of the revolution of the moon4 ,the other inferior planets have still less to show than the earth4 ,venus can barely have more than #bs4 #f;d42 even ,mars' two satellites cannot bring his figure up beyond the slender value of #a;d42 while ,mercury will be amply represented by the smallest coin known at her ,majesty's mint4 ,the same illustration will bring out the contrast between the ,jovian system and our earth system4 ,the rotational share of the former would be totally represented by a sum of nearly #ab1jjj2 of this1 however1 ,jupiter's satellites only contribute about #hi1 notwithstanding that there are four of them4 ,thus ,jupiter's satellites have not one hundredth part of the moment of momentum which the rotation of ,jupiter exhibits4 ,how wide is the contrast between this state of things and the earth-moon system1 for the earth does not contain in its rotation one-fifth of the moment of momentum that the moon has in its revolution2 in fact1 the moon has gradually robbed the earth1 which originally possessed #ais4 #e;d41 of which the moon has carried off all but #cs4 #d;d4 ,and this process is still going on1 so that ultimately the earth will be left very poor1 though not absolutely penniless1 at least if the retention of a halfpenny can be regarded as justifying that assertion4 ,saturn1 revolving as it does with great rapidity1 and having a very large mass1 possesses about #bgjj1 while ,uranus and ,neptune taken together would figure for about the same amount4 ,in conclusion1 let us revert again to the two critical conditions of the earth-moon system4 ,as to what happened before the first critical period1 the tides tell us nothing1 and every other line of reasoning very little2 we can to some extent foresee what may happen after the second critical epoch is reached1 at a time so remote that ,i do not venture even to express the number of ciphers which ought to follow the significant digit in the expression for the number of years4 ,i mentioned1 however1 that at this time the sun tides will produce the effect of applying a still further brake to the rotation of the earth1 so that ultimately the month will have become a shorter period than the day4 ,it is therefore interesting for us to trace out the tidal history of a system in which the satellite revolves around the primary in less time than the primary takes to go round on its own axis--such a system1 in fact1 as ,mars would present at this moment were the outer satellite to be abstracted4 ,the effect of the tides on the planet raised by its satellite would then be to accelerate its rotation2 for as the planet1 so to speak1 lags behind the tides1 friction would now manifest itself by the continuous endeavour to drag the primary round faster4 ,the gain of speed1 however1 thus attained would involve the primary in performing more than its original share of the moment of momentum2 less moment of momentum would therefore remain to be done by the satellite1 and the only way to accomplish this would be for the satellite to come inwards and revolve in a smaller orbit4 ,we might indeed have inferred this from the considerations of energy alone1 for whatever happens in the deformation of the orbit1 heat is produced by the friction1 and this heat is lost1 and the total energy of the system must consequently decline4 ,now if it be a consequence of the tides that the velocity of the primary is accelerated1 the energy corresponding to that velocity is also increased4 ,hence the primary has more energy than it had before2 this energy must have been obtained at the expense of the satellite2 the satellite must therefore draw inwards until it has yielded up enough of energy not alone to account for the increased energy of the primary1 but also for the absolute loss of energy by which the whole operation is characterized4 ,it therefore appears that in the excessively remote future the retreat of the moon will not only be checked1 but that the moon may actually return to a point to be determined by the changes in the earth's rotation4 ,it is1 however1 extremely difficult to follow up the study of a case where the problem of three bodies has become even more complicated than usual4 ,the importance of tidal evolution in our solar system has also to be viewed in connection with the celebrated nebular hypothesis of the origin of the solar system4 ,of course it would be understood that tidal evolution is in no sense a rival doctrine to that of the nebular theory4 ,the nebular origin of the sun and the planets sculptured out the main features of our system2 tidal evolution has merely come into play as a subsidiary agent1 by which a detail here or a feature there has been chiselled into perfect form4 ,in the nebular theory it is believed that the planets and the sun have all originated from the cooling and the contraction of a mighty heated mass of vapours4 ,of late years this theory1 in its main outlines at all events1 has strengthened its hold on the belief of those who try to interpret nature in the past by what we see in the present4 ,the fact that our system at present contains some heat in other bodies as well as in the sun1 and the fact that the laws of heat require continual loss by radiation1 demonstrate that our system1 if we look back far enough1 and if the present laws have acted1 must have had in part1 at all events1 an origin like that which the nebular theory would suppose4 ,i feel that ,i have in the progress of these two lectures been only able to give the merest outline of the theory of tidal evolution in its application to the earth-moon system4 ,indeed ,i have been obliged1 by the nature of the subject1 to omit almost entirely any reference to a large body of the parts of the theory4 ,i cannot bring myself to close these lectures without just alluding to this omission1 and without giving expression to the fact1 that ,i feel it is impossible for me to have rendered adequate justice to the strength of the argument on which we claim that tidal evolution is the most rational mode of accounting for the present condition in which we find the earth-moon system4 ,of course it will be understood that we have never contended that the tides offer the only conceivable theory as to the present condition of things4 ,the argument lies in this wise4 ,a certain body of facts are patent to our observation4 ,the tides offer an explanation as to the origin of these facts4 ,the tides are a vera causa1 and in the absence of other suggested causes1 the tidal theory holds the field4 ,but much will depend on the volume and the significance of the group of associated facts of which the doctrine offers a solution4 ,the facts that it has been in my power to discuss within the compass of discourses like the present1 only give a very meagre and inadequate notion of the entire phenomena connected with the moon which the tides will explain4 ,we have not unfrequently1 for the sake of simplicity1 spoken of the moon's orbit as circular1 and we have not even alluded to the fact that the plane of that orbit is inclined to the ecliptic4 ,a comprehensive theory of the moon's origin should render an account of the eccentricity of the moon's orbit2 it must also involve the obliquity of the ecliptic1 the inclination of the moon's orbit1 and the direction of the moon's axis4 ,i have been perforce compelled to omit the discussion of these attributes of the earth-moon system1 and in doing so ,i have inflicted what is really an injustice on the tidal theory4 ,for it is the chief claim of the theory of tidal evolution1 as expounded by ,professor ,darwin1 that it links together all these various features of the earth-moon system4 ,it affords a connected explanation1 not only of the fact that the moon always turns the same face to the earth1 but also of the eccentricity of the moon's path around the earth1 and the still more difficult points about the inclinations of the various axes and orbits of the planets4 ,it is the consideration of these points that forms the stronghold of the doctrine of tidal evolution4 ,for when we find that a theory depending upon influences that undoubtedly exist1 and are in ceaseless action around us1 can at the same time bring into connection and offer a common explanation of a number of phenomena which would otherwise have no common bond of union1 it is impossible to refuse to believe that such a theory does actually correspond to nature4 ,the greatest of mathematicians have ever found in astronomy problems which tax1 and problems which greatly surpass1 the utmost efforts of which they are capable4 ,the usual way in which the powers of the mathematician have been awakened into action is by the effort to remove some glaring discrepancy between an imperfect theory and the facts of observation4 ,the genius of a ,laplace or a ,lagrange was expended1 and worthily expended1 in efforts to show how one planet acted on another planet1 and produced irregularities in its orbit2 the genius of an ,adams and a ,leverrier was nobly applied to explain the irregularities in the motion of ,uranus1 and to discover a cause of those irregularities in the unseen ,neptune4 ,in all these cases1 and in many others which might be mentioned1 the mathematician has been stimulated by the laudable anxiety to clear away some blemish from the theory of gravitation throughout the system4 ,the blemish was seen to exist before its removal was suggested4 ,in that application of mathematics with which we have been concerned in these lectures the call for the mathematician has been of quite a different kind4 ,a certain familiar phenomenon on our sea-coasts has invited attention4 ,the tidal ripples murmur a secret1 but not for every ear4 ,to interpret that secret fully1 the hearer must be a mathematician4 ,even then the interpretation can only be won after the profoundest efforts of thought and attention1 but at last the language has been made intelligible4 ,the labour has been gloriously rewarded1 and an interesting chapter of our earth's history has for the first time been written4 ,in the progress of these lectures ,i have sought to interest you in those profound investigations which the modern mathematician has made in his efforts to explore the secrets of nature4 ,he has felt that the laws of motion1 as we understand them1 are bounded by no considerations of space1 are limited by no duration of time1 and he has commenced to speculate on the logical consequences of those laws when time of indefinite duration is assumed to be at his disposal4 ,from the very nature of the case1 observations for confirmation were impossible4 ,phenomena that required millions of years for their development cannot be submitted to the instruments in our observatories4 ,but this is perhaps one of the special reasons which make such investigations of peculiar interest1 and entitle us to speak of the revelations of ,time and ,tide as a romance of modern science4 ,,index4 ,aberdeen1 tides at1 #bc ,action and reaction1 #fi ,adams1 discoverer of ,neptune1 #ahf ,admiralty ,manual of ,scientific ,inquiry1 #cj1 #ca1 #cd ,admiralty tide tables1 #bi ,africa1 ,krakatoa dust over1 #hd ,analysis1 harmonic1 of tides1 #cd ,analysis of tide into its constituents1 #cc ,ancient tides on moon1 #acf ,annus ,magnus of solar system1 #gb ,areas1 conservation of1 #fa ,arklow1 tides at1 #bc ,ascension1 tides at1 #bd ,astronomical chronology1 #adg ,atlantic1 ,krakatoa dust over1 #hd ,atlantic1 tides in1 #ch ,atmosphere1 tides in1 #dj ,avon1 tides in1 at ,bristol1 #ci ,ball-room1 illustration1 #fb ,barometric records of ,krakatoa airwave1 #hb ,batavia1 ,krakatoa heard at1 #hc ,bath1 hot waters at1 #hi ,bay of ,fundy1 #ci ,beds of rock1 how formed1 #adi ,birmingham1 lecture at1 #adh ,birth of moon1 #aah ,blast iron furnace1 #hg ,blue sun produced by ,krakatoa1 #hd ,bodily tides of moon1 #aca ,brake illustrating friction1 #fh ,brickwork as a non-conductor1 #hg ,bristol ,channel1 #ci ,bucket of water1 oscillations in1 #ih ,cambrian rocks1 #aec ,cannon-ball1 energy of1 #ea ,cardiff1 tides at1 #ci ,caspian ,sea1 tides in1 #ch ,casting1 cooling of1 #gi ,celebes1 ,krakatoa heard at1 #hc ,central ,america1 #hf ,change and full1 tides at1 #bb ,chemical action in earth1 #ha ,chepstow1 tides at1 #ci ,chronology1 the two systems of1 #adg ,clifton1 tides in ,avon at1 #ci ,clock1 illustration of1 #di ,coal1 #aa ,coincidence of moon's rotation and revolution1 #acd ,combustion1 heat of1 #ib ,cones1 volume of1 #aea ,conservation of spin or areas1 #fd ,constituent tides1 #ce ,consumption of energy by tides1 #dh ,cooling1 laws of1 #gi ,cooling of earth from primitive heat1 #hg ,crane1 brakes on1 #fi ,craters of moon1 #id ,critical epoch in earth's history1 #ge ,darwin1 #ab ,darwin1 ,g4 ,h41 in ,admiralty ,manual1 #cd ,darwin1 ,g4 ,h41 on tidal evolution1 #ig1 #aah1 #abb1 #aea1 #ahe ,day and month equal1 #ajg ,day at present increasing1 #fh ,day of #adjj hours1 #aba ,day of #c or #d hours1 #ge ,decline of earth's heat1 #ij ,diagram showing why moon recedes1 #gj ,diego ,garcia1 ,krakatoa heard at1 #hc ,difficulties of tidal evolution1 #dh ,dives and ,lazarus1 #afa ,dust clouds1 #hd ,dynamical principle1 #fe ,dynamical stability1 #aaj ,earth and moon as rigid body1 #ajg ,earth a fly-wheel1 #eg ,earth1 fusion of1 #ab ,earth1 heat of1 #hj ,earth in highly heated early state1 #ib ,earth's crust1 ,lyell on1 #ab ,earth's history1 #ab ,earth-moon system originally1 #ajg ,earthquakes1 #hj ,eccentricity of moon's path explained1 #ahd ,ecliptic1 obliquity of1 explained1 #ahd ,economic aspects of tides1 #de ,egg on end1 #ajh ,elliptic orbit of moon1 #aac ,endowment of moment of momentum1 #afg ,energy for tides1 whence1 #fg ,energy lost by tides1 #fj ,energy of moon's position1 #ej ,energy of motion1 #ea ,energy of separation1 #ej ,energy1 sources of1 #di ,equality of day and month1 #abb ,equilibrium1 stable and unstable1 #ajh ,equinoxes1 precession of1 #gb ,eruption of volcanoes1 #hj ,establishment1 #cj ,estuary1 tides in1 #aeb ,explosion of ,krakatoa1 #hc ,extinct season1 moon1 #abi ,fiji1 #bd ,fishes1 periods of fossil1 #ab ,fitzgerald1 ,prof41 on ,hertz' experiments1 #aja ,floating log1 #aji ,fortnightly tide1 #cd ,friction brake1 #fi ,friction1 tidal1 #dj1 #dg ,full and change1 tides during1 #bb ,fundy1 ,bay of1 tides in1 #ci ,gauge for tides1 #ca ,geological chronology1 #adf ,geology and tides1 #add ,geometric series1 #ib ,geysers1 #he ,gibraltar1 ,straits of1 tides in1 #ch ,glimpse through the corridors of time1 #adh ,glorious sunsets from ,krakatoa1 #he ,greatest length of day1 #aba ,greatest length of month1 #abb ,greatest tides1 #ci ,green moons from ,krakatoa dust1 #hd ,greenock1 tides at1 #bb ,grindstone1 rupture of1 #gf ,gunpowder1 energy from1 #ea ,hall1 ,prof4 ,a41 discovers satellites of ,mars1 #adb ,harmonic analyzer1 #ce ,heated body1 cooling of1 #gi ,height of tide1 how to measure1 #bi ,helmholtz explains the constant face of the moon1 #acf ,hertz1 undulations of ether1 #aja ,high water1 is it under the moon8 #ba ,high water1 simple rules for1 #bf ,high water twice a day1 #ah ,hot springs1 #he ,hot waters of ,bath1 #hi ,hour of high water1 how found1 #bf ,howth1 #be ,hull1 tides at1 #bc ,ichthyosaurus1 #adh ,impulse1 effect of timed1 #aja ,incandescence of earth's interior1 #he ,incandescence of moon1 #id ,indian ocean1 #hd ,initial condition of earth and moon1 #gf ,instability1 dynamical1 #aaj ,interval1 luni-tidal1 #cj ,iron1 machine for punching1 #ee ,iron smelting1 #hg ,java1 #hb ,jupiter1 #ada1 #aef ,kepler's ,law1 #ec1 #aad ,kerguelen ,island1 #bd ,kingstown1 #ba ,krakatoa1 #eh ,lagrange1 #ahf ,language1 origin of1 #ab ,laplace1 #ahf ,laurentian rocks1 #adh ,law of tides1 #ae ,lazarus and ,dives1 #afa ,leakage of heat1 #hf ,length of day increasing1 #fh ,leverrier discovers ,neptune1 #ahf ,london1 tides at1 #bc1 #bg ,lunar and solar tides compared1 #ae ,lunar diurnal tide1 #cd ,luni-tidal interval1 #cj ,lyell1 #ab ,macassar1 explosion of ,krakatoa heard at1 #hc ,machine for analyzing tides1 #ce ,machine for predicting tides1 #ce ,machine for observing tides1 #cb ,malahide estuary1 #aeb ,mars1 moons of1 #adb ,mass of moon found from tides1 #af ,max ,mller1 #ab ,maximum and minimum1 #ajd ,mediterranean1 tides in1 #ch ,meridian position of moon at high water1 #bf ,mills1 tidal1 #dc ,mines1 heat in1 #hf ,moment of momentum1 how estimated1 #agj ,month equals day1 #abb ,month of #adjj hours1 #abb ,month with greatest number of days1 #aah ,moon and tides connected1 #ac ,moon1 constant face of1 #acf ,moon-energy1 #ed ,moon1 volcanic activity on1 #id ,moons of ,mars1 #adb ,motion1 perpetual1 #dh ,nautical almanac1 #bf ,neap tides1 #af ,needles1 tides at1 #bc ,niagara1 utilization of1 #dg ,noise from ,krakatoa1 #hc ,numerical data of tidal evolution1 #abg ,obliquity of ecliptic and tides1 #ahd ,observations of tides1 #bi ,ocean tides1 #ch ,orbit of earth1 changes in1 #gb ,orbit of moon explained by tides1 #ahd ,origin of moon1 #ajj ,oscillations of water1 #ih ,pacific1 #hd ,paradox of geometric series1 #ib ,pendulum1 motion of1 #di ,periodic phenomena1 #ga ,periods of rotation and revolution of moon equal1 #dc ,periods of tides1 #cd ,perpetual motion1 #dh ,planets and tides1 #aee ,precession of equinoxes1 #gb ,prediction of tides1 #ce ,principal tides1 #cc ,punching-engine1 #ee ,purser1 ,professor1 #fa ,railway brakes1 #fh ,reaction and action1 #fi ,relative rotation1 #ajd ,relative tides on earth and moon1 #aca ,reptiles1 fossil1 #aa ,retreat of moon explained1 #fi ,retrospect of moon's history1 #gg ,rhine1 mills on1 #dc ,ring theory of moon's origin1 #if ,rivers1 #adi ,rocks1 formation of1 #aej ,rodriguez1 ,krakatoa heard at1 #hc ,rolling mills1 #ef ,rotation of moon on its axis1 #acc ,rupture of earth1 #gf ,rupture of grindstone1 #ge ,st4 ,helena1 tides at1 #ch ,santa ,cruz1 #bd ,satellites of ,jupiter1 #ace ,saturn1 #ada ,seasons1 #gb ,seas1 so called1 on moon1 #abi ,secondary ages1 #aa ,sligo1 tides at1 #bc ,small tides1 #ch ,smelting of iron1 #hg ,solar and lunar tides compared1 #ae ,solar system1 #aee ,solar tides1 #ad ,solar tides1 ultimate importance of1 #aci ,sounds from ,krakatoa1 #hc ,source of tidal energy1 #fg ,spring tides1 #af ,stability1 dynamical1 #aaj ,stable equilibrium1 #ajh ,stages of special importance1 #aag ,stalactites1 #gc ,steam engine and tides compared1 #de ,strata1 formation of1 aided by tides1 #aed ,sumatra1 #hb ,sun1 rotation of1 #acd ,sunbeams stored1 #aa ,sunda1 #hb ,sunsets1 ,krakatoa1 #he ,table of luni-tidal corrections1 #cj ,telescope at ,washington1 #adc ,temperature of space1 #ia ,tertiary ages1 #aa ,thames1 tides in1 #abe ,thomson1 ,sir ,w41 #cf1 #dd ,tidal currents used in making rocks1 #aec ,tidal efficiency1 how estimated1 #ad ,tidal evolution1 to whom due1 #ig ,tide gauge1 #ca ,tide mills1 #dd ,tide predicting engine1 #cf ,tides and geology1 #add ,tides at ,london bridge1 #bh ,tides1 change and full1 #bb ,tides1 commercial value of1 #dg ,tides1 due to moon1 #ac ,tides1 how to observe1 #bi ,tides in ,jupiter1 #ach ,tides in moon1 #abh ,tides1 small1 #ch ,tides1 solar and lunar compared1 #ae ,tides varying inversely as cube of distance1 #ae ,time and tide1 #ab ,timor1 ,krakatoa heard at1 #hc ,tralee1 tides at1 #bc ,tyndall1 #gc ,tynemouth1 #bc ,undulations of a fluid globe1 #ii ,undulations of air from ,krakatoa1 #hb ,unintermitting phenomena1 #ga ,unstable equilibrium1 #ajh ,uranus1 #adb ,vast tides1 #ci ,volcanoes1 #hj ,voyage of ,krakatoa dust1 #hd ,washington reflector1 #adb ,water absent from moon1 #abi ,water-wheels under ,london bridge1 #dc ,waves of air from ,krakatoa1 #hb ,whewell on tides1 #cj ,work done by tides1 #db1 #dg ,work of tides1 #db ,yarmouth1 tides at1 #bc ,,the ,,end4 ,richard ,clay ,sons1 ,limited1 ,london ,bungay4