THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA PRESENTED BY PROF. CHARLES A. KOFOID AND MRS. PRUDENCE W. KOFOID AN ESSAY THE TIDES, AN ESSAY THE TIDES THEORY OF THE TWO FORCES ALEXANDER WILCOCKS, M. D. PHILADELPHIA: PARRY & MCMILLAN, PUBLISHERS, ST-CCESSORS TO A. HART, (LATE CAREY & HART.) 1855. Entered according to the Act of Congress, in the year 185/5, by ALEXANDER WILCOCKS, M. D., In the Clerk's Office of the District Court of the Eastern District o Pennsylvania. VIS TO PROFESSOR J. K. MITCHELL, TllKSi; PAGKS ARE GRATEFULLY INSCRIBED BY HIS FRIENP ANT) FORMER PUPIL, THE AUTHOR. I M363107 MEMORANDUM. CERTAIN notes which appear in the Appendix were made upon this paper, when in manuscript, by a " mathematical friend" of the editors of a scientific journal in New England. The notes were intended to be conclu- sive against the truth of the paper : a perusal of them, as well as of the replies to them, will enable one to form an opinion as to their having accomplished their aim. They are of value, because they point out the respects in which the Newtonian theory differs from the views here advocated, and have enabled the author to fortify those points with additional argument. For these reasons they are published. INTRODUCTION. THE theory of the tides, which is now generally received, is the result of the speculations of Sir Isaac Newton, upon which have been engrafted some modifications by philosophers who have lived since his day. The following quotation, from Brande's Dic- tionary, is a concise definition of the subject : " TIDES. The alternate rise and fall of the waters of the ocean. The Moon is the principal agent in the production of the tides ; but they are modified with respect to their height, and the times at which they happen, by the action of the Sun. The effect of the planets is inappreciable. " The attractive force of a body on a distant particle of matter, varying inversely as the square of the distance, the particles of the earth on the side next the moon will be attracted with 10 INTRODUCTION. a greater, and those on the opposite side with a smaller force, than those which are situated intermediately. The gravitation towards the earth's centre of the particles nearest the moon will, therefore, be diminished, and consequently, if at liberty to move among themselves, they will rise abve the general level. In like man- ner, the moon's attraction on the most distant particles being less than on the central ones, their relative gravitation towards the centre will also be diminished, and the water will conse- quently be heaped up on the side of the earth which is turned away from the moon. Hence, if the earth were at rest, the ocean would take the form of an oblong spheroid, with its longer axis passing through the attracting body ; and it may be shown from theory, that the spheroid would be in equilibrium under the influence of the moon's attraction, if the longer semi-axis exceeded the shorter by fifty-eight inches. But in consequence of the rapid rotation of the earth about its axis, the spheroid of equilibrium is never fully formed ; for, before the waters can take their level, the vertex of the spheroid has shifted its position on INTRODUCTION. 11 the earth's surface, in consequence of which an immensely broad and very flat wave is formed, which follows the motion of the moon at some interval of time. In the open sea, the time of high water is, in general, from two to three hours after the moon's transit over the meridian, either above or below the horizon. The tidal wave, it is to be observed, is entirely different from a cur- rent, the particles of water merely rise and fall ; but except when the wave passes over shallows, or approaches the shore, there is little or no pro- gressive motion." Dr. Lardner, in his "Lectures on Science and Art," makes use of the following figure to illus- trate the moon's influence in the causation of tides, and thus describes the phenomenon : "LetABCDEF p G H represent the globe of the earth, and to simplify the explanation, let us first suppose the en- tire surface of the globe to be covered 12 INTRODUCTION. with water. Let M, the moon, be placed at the distance K L from the nearest point of the surface of the earth. Now, it will be very apparent that the various points of the earth's surface are at different distances from the moon, M. A and G are more remote than H. B F still more remote. C and E more distant again ; and D more remote than all. The attraction which the moon exercises at H, is therefore greater than that which it exercises at A arid G, and still greater than that . which it produces at B and F ; and the attraction which it exercises at D, is least of all. Now, this attraction equally affects matter in every state and condition. It affects the particles of fluid as well as solid matter, but there is this difference between these effects, that where it acts upon solid matter, the component parts of which are at different dis- tances from it, and therefore subject to different attractions, it will not disturb the relative arrange- ment of these particles, since such disturbances or disarrangements are prevented by the cohesion which characterizes a solid body ; but this is not the case with fluid, the particles of which are mobile, and which, when solicited by different INTRODUCTION. 13 forces, will have their relative arrrangements disturbed in a corresponding manner. " The attraction which the moon exercises upon the shell of water which is collected immediately under it near the point Z, is greater than that \vhich it exercises upon the solid mass of the globe at H and D ; consequently, there will be a greater tendency of this attraction to draw the fluid which rests upon the surface at H towards the moon, than to draw the solid mass of the earth, which is more distant. "As the fluid, by its nature, is free to obey this excess of attraction, it will necessarily heap itself up in a pile or wave, at H, forming a more convex protuberance, as represented in the figure between R and I. Thus, high water will take place at H, immediately under the moon. The water which thus collects at H, will necessarily flow from the regions B and F, whence, therefore, there will be a diminished quantity of water in the same proportion. " But let us now consider what happens to that part of the earth, D, most remote from the moon. Here the waters being more remote from the moon 14 INTRODUCTION. than the solid mass of the earth under them, will be less attracted, and consequently, will have a less tendency to gravitate towards the moon. The solid mass of the earth, D H, will, as it were, recede from the waters at N, in virtue of the excess of attraction, leaving these waters behind it, which will thus be heaped up at N, so as to form a convex protuberance between L and K, similar exactly to that which we have already described between R and I. As the difference between the attraction of the moon on the waters at Z, and the solid earth under the waters, is nearly the same as the difference between its attraction on the latter and upon the waters at N, it follows that the height of the fluid protuberances at Z and N are equal. In other words, the height of the tide on opposite sides of the earth, the one being under the moon, and the other most remote from it, are equal." INTRODUCTION. 15 Whoever will examine this theory, with a determination to admit nothing that is not either self-evident or demonstrated, will discover that it is insufficient to account for one of the most striking phenomena of the tides. We are informed that the particles of water on the side of the earth next the moon are attracted with a greater force than those which are more remote. That the gravitation towards the centre of the earth of those particles will therefore be diminished, and that consequently, they will rise above the general level.* All this is perfectly reasonable. But we are also informed that the moon's attraction on the most distant particles being less than on the central ones, their relative gravitation towards the centre will also be diminished, and that the waters will consequently be heaped up on the side of the earth which is turned away from the moon.f This is the grand fallacy. A careful study of the influence of the forces under consideration, * Brande's Dictionary. t Ibidem. 16 INTRODUCTION. upon the water at N, must convince any one of the following facts : 1st. That the water at N is, by its own gravity, attracted in the direction of the earth's centre, N O. 2d. That whatever attractive influence the moon can exert upon the water at N, must be to draw it in the direction of her own mass, N M, which is coincident with N O. Here then are two forces tending to op- pose a rise of the wa- ters at N.* It is undoubtedly true that the attractive force of the moon upon the waters at N is less than upon the solid mass of the earth at D H, by reason of the greater distance of N ; but it is also true that what attraction the moon can exert at N, being in the same direction as the earth's See Appendix, Note 1st. INTRODUCTION. 17 attraction, must tend to force the waters below their general level. There can be no greater absurdity than the idea of the effect of a force being reversed,* merely because its intensity is diminished.! * See Appendix, Note 2d. t " There can be no greater absurdity than the idea of the effect of a force being reversed, merely because its in- tensity is diminished." One difficulty in the way of appreciating the truth of the above sentence, exists in the indefinite meaning attached to the word attraction. Ther.e are some persons who regard it as the disposition which two bodies have to approach each other, without any actual lessening of their distance. Others deem the approach of the bodies to be involved in the strict meaning of the term. The former is the correct definition. Now, as regards the moon's action upon the earth's nadir waters, it cannot be denied, that if the body of the earth were removed, and the nadir waters left, the moon's attrac- tion would give a tendency to those waters to approach her mass, which, in the absence of the tangential force, would result in actual approach. This being admitted, and the idea being untenable that the moon's attraction for the earth occasions an actual ap- proach of the earth to the moon, by what influence which the moon can exert upon the body of the earth, can its nadir waters be driven up against the force of their gravity to their parent orb ? and against their gravity to the lunar mass ? 2 18 INTRODUCTION. Dr. Lardner, in treating upon this point, says : " The solid mass of the earth, D H, will, as it were, recede from the waters at N, in virtue of the excess of attraction, leaving these waters behind it, which will thus be heaped up at N, so as to form a convex protuberance between L and K, similar exactly to that which we have already described between R and I." Now, against this recession of the solid mass of the earth, by which its waters are left behind it, it must be urged, that the velocity of falling bodies* upon the surface of the earth is a matter thoroughly understood ; consequently, the rate at which the earth must recede, in order that its waters should not overtake it, is an affair of calculation. Further, as in order to produce a tide when the moon is in the nadir, the recession must be in the direction of that body, calculation would also show in what time the earth and moon would come together. These are some of the inevitable accompani- ments of the theory of the tides now generally received. * See Appendix , Note 3d. INTRODUCTION. 19 The prime error of this theory lies in the fact that its author, with a more perfect knowledge of the relations which subsist between the earth and its satellite, than any other man of his day, did, in his dissertation upon the tides, grossly misstate the case. His views upon the subject appear in Book 1st, Proposition LXVL, Theorem XXVI., of the " Principia," in which he conceives three bodies, whose forces decrease in a duplicate ratio of the distances, to attract each other mutually ; and the accelerative attraction of any two towards the third to be between themselves reciprocally as the squares of the distances ; and the two least to revolve about the greatest. C L In case first, under this proposition, he supposes the body P to be urged with a force towards T, 20 INTRODUCTION. which arises from the mutual attraction of the bodies T and P. By this force alone, he says, the body, P, would describe around the body, T, by the radius, P T, areas proportional to the times, and an ellipsis,* whose focus is in the centre of the body, T. Here is the error, for it is at variance with the laws of nature, for a body to describe an ellipsis, whose focus is in the centre of another body.f Under section XL, Book 1st, of the Principia, the author fully and decidedly expresses his views upon this point ; he says : " Attractions are made towards bodies, and the actions of the bodies attracted and attracting, are .always reciprocal and equal, by Law III. ; so that if there are two bodies, neither the attracted nor the attracting body is truly at rest, but both, (by Cor. IV., of the Laws of Motion,) being, as it were, mutually attracted, revolve about a com- mon centre of gravity. And if there be more bodies, which are either attracted by one single one which is attracted by them again, or which * See Appendix, Note 4th. t See Appendix, Note 5th. INTRODUCTION. 21 all of them, attract each other mutually, these bodies will be so moved among themselves, as that their common centre of gravity will either be at rest, or move uniformly forward in a right line." In Proposition LXVI. and its first seventeen Corollaries, Sir Isaac Newton gives his views of the influence which the three bodies P, S and T would, by their gravity, exert upon each other. In Corollary 18, he says, " By the same laws by which the body P revolves about the body T, let us suppose many fluid bodies to move round T at equal distances from it ; and to be so numerous that they may all become contiguous to each other, so as to form a fluid annulus, or ring of a round figure, and concentrical to the body T; and the several parts of this annulus, performing their motions by the same law as the body P will draw- near to the body T, and move swifter in the con- junction and opposition of themselves and the body S, than in the quadratures. And the nodes of this annulus or its intersections with the plane of the orbit of the body S or T, will rest at the syzygies, but out of the syzygies they will be 22 INTRODUCTION. carried backward, or in antecedentia ; and with the greatest swiftness in the quadratures, and more slowly in other places. The inclination of this annulus also will vary, and its axis will oscillate each revolution, and when the revolution is completed, will return to its former situation, except only that it will be carried round a little by the precession of the nodes." " Cor. 19. Suppose now the spherical body T, consisting of some matter not fluid, to be enlarged, and extend itself on every side as far as that annulus, and that a channel were cut all round its circumference, containing water, and that this sphere revolves uniformly about its own axis in the same periodical time. This water being accelerated and retarded by turns, (as in the last corollary,) will be swifter at the syzygies, and slower at the quadratures, than the surface of the globe, and so will ebb and flow in its channel after the manner of the sea. If the attraction of the body S were taken a\vay, the water would acquire no motion of flux and reflux by revolving round the quiescent centre of the globe. The case is the same of a globe moving INTRODUCTION. 23 uniformly forward in a right line, and in the meantime revolving about its centre, (by Cor. 5, of the Laws of Motion,) and of a globe uniformly attracted from its rectilinear course, (by Cor. 6 of the same laws.) But let the body S come to act upon it, and by its unequable attraction, the water will receive this new motion ; for there will be a stronger attraction upon that part of the water which is nearest to the body, and a weaker upon that part which is more remote. And the force L M will attract the water downwards at the quadratures, and depress it as far as the syzygies, and the force KL will attract it up- ward in the syzygies, and withhold its descent, and make it rise as far as the quadratures, except only in so far as the motion of flux and reflux may be directed by the channel of the water, and be a little retarded by friction." This quotation, from the Principia, is made to prove that the views of its author, concerning the cause of tides, were similar to those now generally entertained; and also, more particularly to show that in Corollaries 18 and 19 of Proposition LXVL, in which he applies the laws which govern 24 INTRODUCTION. the three bodies P, S and T, to the explanation of the flux and reflux of the sea ; and in which the bodies T and S are intended to represent the earth and moon, not the slightest allusion is made to the fact that they revolve about a centre common to the two.* Nineteen years before Sir Isaac Newton pre- sented to the Royal Society his " Philosophic Naturalis Principia Mathematica," viz., in 1666, Dr. Wallis, in letters to Mr. Boyle, attributed the alternate rise and depression, of the ocean to the consideration, that the common centre of gravity of the moon and the earth describes an orbit about the sun, while they revolve about this common centre. Of Dr. Wallis's letters, Dr. Milner says, " His language evinces great sagacity, but it is the language of surmise." It is remarkable that Sir Isaac Newton, in his exposition of the tides, should not have availed himself of the ingenious truth suggested by Dr. a truth which could not be demonstrated * See Appendix, Note 6th. t See Appendix, Note 7th. INTRODUCTION. 25 by the mind which conceived it, for want of data, which the Principia afterwards supplied.* The article on tides in the Encyclopaedia Bri- tannica, describes the earth as " constantly falling towards the moon from a tangent to the circle it describes round their common centre of gravity." These few words imply the correct idea of the true cause of the tides ; and it is extraordinary that the authors of the more modern treatises upon the subject should not have been alive to their importance. But these words, important as they are in the explanation of tidal phenomena, must not receive too general an interpretation, for there are parts of the earth which, in the revolution round the common centre of gravity of the earth and moon, move from the tangent to their orbit in a direction from the moon. This is true of those parts of the earth which are nearer the moon than the common centre of gravity is. * See Appendix, Note 8th. 26 INTRODUCTION. A B is a small arc, representing a portion of the earth's surface ; C is the common centre of gravity of the earth and moon ; C M is the direction of the moon ; and Z is that point of INTRODUCTION. 27 the earth's surface where the moon is in the zenith. The clotted curved line z Z z is a part of the monthly orbit of that point of the earth's surface where the moon is in the zenith. TZ is a tangent to that orbit. As the earth and moon advance in their orbits, the latter will arrive at a point in the direction CL, and the part of the earth's surface to which she is vertical will reach the position Y, and will thereby deviate from the tangent to its orbit, by the amount and in the direction of x Y, which is in the direction from the moon, Q E D. An effort has now been made to show the errors and inconsistencies of the views which have been entertained upon the subject of the tides. For the purpose of affording an explanation of some of the phenomena which have hitherto been without it, an offer is made of the following theory :* * See Appendix, Note 9th. THEORY OF THE TWO FORCES. THE astronomical conditions upon which the cause and variations of the tides depend, are the following : The earth rotates upon its axis once in a day. The earth and the moon revolve about their common centre of gravity once in a month. The earth and moon, as an independent system, revolve about the common centre of the solar system once in a year. The plane of the earth's equator lies at an angle of 23 28' to that of the ecliptic. The plane of the moon's orbit lies at an angle of 5 9' to that of the ecliptic. The mean distance of the moon from the earth is 60 semi-diameters of the latter. The greatest and least distances of the earth and moon asunder, are as 64 to 56. The mean distance of the earth from the sun is THEORY OF THE TWO FORCES. 29 23.750 semi-diameters of the former, or 400 semi- axes of the moon's orbit. The greatest and least distances of the sun and earth asunder, are as 100 to 97. The mass of the moon, compared to. that of the earth, is as 1 to 75. The mass of the sun, compared to that of the earth, is as 354.936 to 1. Certain laws pertaining to matter in every form, have a special application to the subject of the tides. * Every particle of ponderable matter in the universe, attracts every other particle The bodies of space attract each other with a force directly as their masses, and inversely pro- portional to the squares of their distances. With a given angular velocity, the centrifugal force of a revolving body is as the radius of its orbit. The common centre of gravity of two bodies divides the line joining their centres into two parts, which are reciprocally as the bodies. The common centre of gravity of two or more bodies acting upon each other, may either be at 30 THEORY OF THE TWO FORCES. rest, or move in a right line ; or, the system of bodies being uniformly attracted from a rectilinear course, their common centre may move in a curve. The principles which have been here laid down, are so well established, that it is believed no one who is instructed in such matters, will dispute them. It is proposed to show by these principles, the necessity for a periodical flux and reflux of the sea, and to point out a cause for a variation in the height of two successive tides which has not been heretofore noticed. If the earth were at rest, and composed of a homogeneous fluid, its figure would be determined by the gravity of its particles. It would be a perfect sphere, for that admits of a closer approxi- mation of each particle to all the rest than any other form. The rotation of the earth upon its axis has a perturbing influence upon its sphericity ; it gives a tendency to the particles about the equator, to recede from the axis of rotation. This is owing to the centrifugal force created by the rotary THEORY OF THE TWO FORCES. 31 motion, and the form which the earth assumes under the joint influence of gravity, and the cen- trifugal force will be a spheroid; the polar diameter being the shorter. The gravity of the earth's particles, and the centrifugal force created by rotation being causes which undergo no change, the form which the earth takes under their influence, is unvarying. The other motions which the earth has, to- gether with the gravitating influence of the moon and sun, have an effect upon its waters. But, as these causes are varying in their intensity and direction, the form which the waters assume under their influence, is constantly changing. The results of the variable perturbations are tides, and their study is made "more easy by the separate consideration of each set of causes. 32 THEORY OF THE TWO FORCES. LUNAR INFLUENCE. THE earth and the moon revolve every month about a point, which is situated in a right line passing through their centres. This point is their common centre of gravity ; in order to distinguish it from other centres with which it might be con- founded, and to define its relations to the earth and moon, it will be referred to as the Geoselenic centre; it is within the body of the primary planet. These bodies are by their vis insita impelled in the direction of tangents to their orbits, from which rectilinear course, their mutual attraction uniformly deflects them to the curved paths in which they revolve. The resistance which they offer to their attrac- tion in deflecting them from a straight course, is the centrifugal force, and the attraction which overcomes this, is the centripetal force. LUNAR INFLUENCE. 33 These, the central forces, are in antagonism ; not only is their influence exerted upon the masses of the earth and moon as units, but every atom in each, according to its position, either enjoys an equilibrium, or feels a preponderance of one or the other of the contending forces. Let the larger globe in the figure represent the earth, and the smaller the moon, performing their revolutions about the geoselenic centre, C. The curved line e E e is a portion of the path of the earth's centre in its monthly orbit ; and the curved line mMm is a portion of the moon's path in her monthly orbit. C E is the radius of the earth's monthly orbit. CM is that of the moon. With the radius C E, the centrifugal force of the earth equals the mutual attraction of the earth and moon at the distance E M ; for if the centrifugal force were in excess, the bodies would recede from each other ; and if the attraction were in excess, they would approach ; thus the stability of the system asserts the truth of the proposition. Since the centrifugal force of the earth with the radius C E equals the attraction of the bodies at 3 34 THEORY OF THE TWO FORCES. LUNAR INFLUENCE. 35 the distance EM, the measure of each of the central forces at E may be taken as unity. Experiment has proved that with a given angular velocity the centrifugal force of a revolv- ing body is as the radius of its orbit. And observation has shown, that the attraction of masses varies inversely as the squares of the distance of the attracting bodies. A little attention to the figure will make it very clear, that in the monthly circuit the parts of the earth near N are whirled in an orbit of larger radius than those less distant from the geoselenic centre C ; and also, that the part near N being most distant from the moon, must ex- perience a comparative diminution of her attrac- tive influence. Hence, if we imagine the earth's surface covered with a universal ocean, there will be a rise of the waters at N, (not merely from a diminution of the moon's attraction,) but from the absolute preponderance of the centrifugal force over the diminished centripetency. The parts of the earth which are nearest the moon, are in consequence, most powerfully at- tracted by her : hence, upon the supposition of an 36 THEORY OF THE TWO FORCES. universal ocean, there should be a rise of the waters where the moon is in the zenith. This explanation of the zenith tide has hitherto been deemed sufficient to account for the phenomenon ; but, though the moon's relatively increased attraction is its principal cause, yet it is not the sole cause. All parts of the earth perform the monthly circuit round the geoselenic centre, the point Z in the figure is the nearest part of the earth to the moon ; it is also the nearest point of the earth's surface to the geoselenic centre ; it has therefore the smallest radius to its monthly orbit ; and, consequently, the smallest measure of cen- trifugal force ; but this force, whatever its effi- ciency may be, as it is exerted in the direction of the moon, is auxiliary to her attraction in the production of the zenith tide. Thus it appears that two causes are concerned in the production of each of the tides, viz., the centripetal and the centrifugal forces ; it is also manifest that their relative action in the two cases is widely different for in the zenith tide the two forces conspire to a rise of the waters: whereas, in the nadir tide, the moon's attraction LUNAR INFLUENCE. 37 tending towards a depression of the waters, they only rise from the augmented measure of the centrifugal force. It can be shown by calculating the value of the central forces on those points of the earth's surface, which are the nearest to, and the most remote from the moon, that their influence in causing a rise of the waters upon opposite sides is nearly equal. It has been proved that the value of the central forces at the earth's centre is the same ; and it has been proposed to estimate this at unity. Now, the distance from the earth's centre to the moon's centre is 60 of the earth's semidiameters ; frem the moon's centre to the surface of the earth nearest the moon, 59 semidiameters ; from the moon's centre to the surface of the earth, away from the moon, 61 semidiameters. The squares of these distances being respectively 3.600, 3.481, and 3.721, the attractive influence of the moon upon particles situated on the surface of the earth nearest to her, is to that which she exerts at the earth's centre, as 3.600 is to 3.481, or more simply expressed, as 1.0342 is to 1 ; and 38 THEORY OF THE TWO FORCES. her influence upon the surface most remote from her, to that upon a particle at the earth's centre, is as 3.600 is to 3.721, or as .9674 is to 1. In the adjoining figure, the larger body repre- sents the earth, and the smaller the moon. The arrow shows the direction in which her attraction is exerted, and the numbers on either side show the value of the moon's influence upon those points, while the algebraic signs prefixed to the numbers indicate the position or negative effect of her attraction in causing a rise of the waters. The next point is to determine the position of the geoselenic centre. " The common centre of gravity of two bodies divides the line joining their centres into two parts, which are reciprocally as the bodies." Hutton. Reliable estimates assign to the moon a mass equal to -fg part of the earth; by the above rule, the geoselenic centre must be distant from the earth's centre, ^g part of the whole distance between the earth's centre and the moon's centre. If the distance from the earth's centre to the LUNAR INFLUENCE. 39 40 THEORY OF THE TWO FORCES. moon's centre be 60 of the earth's semidiameters, and the distance from the earth's centre to the common centre be ??Q part of 60 semidiameters ; this latter distance'will be expressed by the frac- tion T 7 9 jj of one semidiameter. This fraction T % of one semidiameter is the radius of the earth's monthly orbit, and measures the unit of the centrifugal force ; it is C E in the figure. Having determined the measure of the radius C E, C Z and C N become known. CZ is 1. .79.21, and C N is l.+.79=1.79. It is necessary to have the measures of C Z and C N in terms of C E the unit. Thus, C E .79 : C Z .21 : : 1. : .2658 and CE .79 : CN 1.79 : : 1. : 2.2658 Having thus obtained the value of each of the central forces at the points Z and N, it remains to show that if the moon's attraction at Z be added to the centrifugal force at this point, and if her attraction at N be deducted from the centrifugal force at that point, the sum of the first, and the difference of the second, will be very nearly the same. LUNAR INFLUENCE. 41 In the above figure, the upper row of numbers represents the measure of the centrifugal force in different parts of the earth, and the arrows its 42 THEORY OF THE TWO FORCES. direction. The second row represents the centri- petal force, and the arrows also the direction. It appears that the sum of the central forces at Z, and their difference at N balance within VCKOUD of a unit. SOLAR INFLUENCE. 43 SOLAR INFLUENCE. THE earth and the moon perform their annual revolution about a point which is common to them and the other bodies of the solar system. From the magnitude of the solar mass, this point, which may be called the systemic centre, is near the body of the sun. Owing to the monthly revolutions of the earth and moon, neither of these bodies describes an exact ellipsis about the systemic centre ; one point in their system does so, and that is their own common centre of gravity. At the geoselenic centre, there exists an equi- librium between the sun.'s attraction and the cen- trifugal force created by the annual revolution of the earth and moon. The proof of this is similar to that which has been adduced to establish an analogous fact relating to the lunar tides, viz., That if the centrifugal force were in excess, the 44 THEORY OF THE TWO FORCES. earth and moon would recede from the sun ; and if the attraction were in excess, they would approach the sun ; as it is known that neither of these changes occurs, it follows of necessity that at the geoselenic centre, the solar central forces must be in equilibrium, and may be estimated at unity. The investigation of the sun's influence in the production of tides develops a complication which does not exist in regard to the moon's influence, viz., that while in the system of the earth and moon, the equilibrium between the central forces is at the earth's centre ; in the system now under consideration, the equilibrium is at the geoselenic centre, which point, owing to the earth's rotation upon its axis, is in relation to the different parts of its surface in constant motion. The influence of this condition in complicating the effect of solar action, can be shown by a figure. Let S be the sun, M the moon, and E the earth; C is the geoselenic centre, the point at which the sun's attraction balances the centrifugal force of the earth and moon, and at which the SOLAR INFLUENCE. 45 46 THEORY OF THE TWO FORCES. .measure of the two central forces is unity. Now, as the point Z is very little nearer the sun than the geoselenic centre C, his attraction must be but slightly greater at Z than at C ; and as the radius S Z is very little less than S C, the centri- fugal force at Z must be also little less than at C. Hence, as the rise of the waters from solar influ- ence is dependent upon the disproportion between the central forces, the sun will have very slight effect in the production of a tide at Z. The reverse of this state of things exists at N. Here the distance from the sun is greater than at C, by nearly the diameter of the globe ; a comparatively great diminution of his attraction must ensue; the radius SN being also much greater than S C, the centrifugal force must receive a corresponding increase. The dispropor- tion between the central forces being great, the sun will have much effect in the production of a tide at N. The inference to be drawn from these facts is, that in the conjunction of the luminaries, the sun has much less power to produce a zenith than a nadir tide. The measure of his influence in the SOLAR INFLUENCE. 47 production of each tide can be shown numeri- cally. The distance between the sun and the geoselenic centre in terms of the earth's semidiameter, is 95 iitf6o 00 = 23 - 750 - The distance between the sun and the point Z is 23.750. .21=23.749.79. The distance between the sun and the point N is 23.750.+1.79:=:23.751.79. The squares of these distances are respectively : 564.062.500. sq. of geoselenic distance. 564.052.525. sq. of zenith distance. 564.147.528. sq. of nadir distance. As the attraction of masses is inversely as the squares of their distance, the sun's influence at Z is to that at C, as 564.062.500. is to 564.052.525., or as 1.000017 is to 1., and his influence at N is to that at C, as 564.062.500. is to 564.147.528., or as .999849 is to 1. ATTRACTION AT Nadir. Geoselenic Centre. Zenith. .999849 1. 1.000017 The distance between the systemic and geosele- nic centres or the radius S C being 23.750. semi- diameters, the radius SZ being 23.749.79 semi- 48 THEORY OF THE TWO FORCES. diameters, and the radius S N being 23.751.79 semidiameters, those numbers are respectively the measures of the centrifugal force at the points C, Z, and N. If the centrifugal force at C 23.750. be unity, the value of that at Z, 23.749.79 will be .999991, and the value of that at N 23.751.99 will be 1.000075 CENTRIFUGAL FORCE AT Nadir. Geoselenic Centre. Zenith. 1.000075 1. .999991 Having now obtained the measure of each of the central forces at the three points, N, C and Z, the value of the sun's influence at these points will be the amount of the disproportion between the central forces. Nadir. Geoselenic Centre. Zenith. Centrif. 1.000075 1. Attraction 1.000017 Attract. .999849 1. Centrif. force .999991 .000226 0. .000026 Thus, when the luminaries are in conjunction, the solar influence upon the nadir tide is to that upon the zenith tide as 226 is to 26, or 8.6 is tol. SOLAR INFLUENCE. 49 When the sun and moon are in opposition, the effect of the condition under consideration upon the solar tides is reversed. Let S be the sun, M the moon, and E the earth, C is the geoselenic centre. The point N being very little farther from the sun than the geoselenic centre, his attrac- tion can be only a little less at N than at C ; and the radius S N being but little longer than S C, the centrifugal force can be only slightly increased. The disproportion between the central forces is in consequence small, and the effect of the sun, in causing a rise of the waters at N, is also small. But the point Z is nearer the sun than the geosele- nic centre is by nearly the diameter of the earth, his attraction must therefore receive a great com- parative increase, and the radius SZ is much shorter than S C, the centrifugal force must con- sequently be much less. There is thus a great disproportion between the central forces at Z, and the sun's influence in producing a tide at that point is correspondingly great. The inference to be drawn from these facts is, that in the opposi- tion of the luminaries, the sun has much more 4 THEORY OF THE TWO FORCES. SOLAR INFLUENCE. 51 influence in the production of the solar zenith tide than in that of the nadir. The measure of his influence in the production of each tide is obtained as follows. The distance from the sun to the geoseleriic centre is 23.750 of the earth's semidiameters. The distance from the sun to the nearest point of the earth's surface when the moon is in opposition, is 23.748.21 semi- diameters ; and from the sun to the most distant point of the earth's surface 23.750.21 semidiame- ters. The squares of these numbers are respectively : 564.062.500. sq. of geoselenic distance. 563.977.478. sq. of zenith distance. 564.072.475. sq. of nadir distance. The sun's attractive influence upon the waters at Z, is to his attraction at C, as 564.062.500. is to 563.977.478., or as 1.000151 is to 1. ; and his attractive influence at N is to his attraction at C, as 564.062.500. is to 564.072.475., or as .999982 is to 1. ATTRACTION AT Nadir. Geoselenic Centre. Zenith, .999982 1. 1.000151 52 THEORY OF THE TWO FORCES. The distance between the systemic and geosele- nic centres, or the radius S C being 23.750. semi- diameters, the radius S Z being, when the sun and moon are in opposition 23.748.21 semidiame- ters, and the radius SN being 23.750.21 semi- diameters, those numbers are respectively the measures of the centrifugal force at the points C, Z and N. If the centrifugal force at C, 23.750. be unity, that at Z 23.748.21 will be .999924, and the value of that at N 23.750.21 will be 1.000009 CENTRIFUGAL FORCE AT Nadir. Geoselenic Centre. Zenith. 1.000009 1. .999924 Having now obtained the measure of each of the central forces at the three points, N, C and Z, the value of the sun's influence at these points will be the amount of the disproportion between the central forces. Nadir. Geoselenic Centre. Zenith. Centrif. 1.000009 1. Attraction 1.000151 Attract. .999982 1. Centrif. force .999924 .000027 0. .000227 SOLAR INFLUENCE. 53 Thus, when the luminaries are in opposition, the sun's influence upon his zenith tide to that upon his nadir tide, is as 227 is to 27, or 8.4 is tol. There are two moments in each lunation when the influence which the sun exerts in the produc- tion of tides on opposite sides of the globe is the same. This occurs when the moon is in the quadratures. . Let S be the sun, M the moon, and E the earth ; the path of the geoselenic centre c C c passes through the earth's centre E, and the central forces are as much in equilibrium at E as they are at C. The distance S Z bears nearly the same proportion to S E, as S E does to S N. The sun's attraction at Z therefore exceeds that at E in nearly the same proportion as his attrac- tion at E exceeds that at N. And the centrifugal force at N exceeds that at E in nearly the same proportion as the centrifugal force at E exceeds that at Z. The inference to be drawn from these facts is, that when the moon is in the quadratures, there is, at those points of the earth's surface where THEORY OF THE TWO FORCES. SOLAR INFLUENCE. 55 the sun is in the zenith and in the nadir, nearly an equal disproportion between the central forces. The measure of his influence upon the waters at each point is as follows : The distance from the sun to the geoselenic cen- tre is 23.750. of the earth's semidiameters. The distance from the sun to the nearest point of the earth's surface when the moon is in the quadra- ture, is 23.749. semidiameters, and from the sun to the farthest point of the earth's surface 23.751. semidiameters. The squares of these numbers are respectively : 564.062.500. sq. of geoselenic distance. 564.015.001. sq. of zenith distance. 564.110.001. sq. of nadir distance. The sun's attractive influence upon the waters at Z is to his attraction at E, as 564.062.500. is to 564.015.001., or as 1.000084 is to 1., and his attractive influence at N is to his attraction at E, as 564.062.500. is to 564.110.001., or as .999916 is to 1. The distance between the systemic and geose- lenic centres or the radius S C being 23.750. semidiameters, the radius S Z being 23.749. semi- 56 THEORY OF THE TWO FORCES, diameters, and the radius SN being 23.751. semi- diameters, those numbers are, when the moon is in the quadratures, the measure of the centrifugal force at the points C, Z and N. If the centrifugal force at C, 23.750., be unity, that at Z, 23.749., will be .999953, and the value of that at N, 23.751., will be 1.000042 Having now obtained the measure of each of the central forces at the points N, C and Z, the value of the sun's influence at these points will be the amount of the disproportion between the central forces. Nadir. Geoselenic Centre. Zenith. Centrif. 1.000042 1. Attraction 1.000084 Attract. .999916 1. Centrif. force .999958 .000126 0. .000126 Thus, when the moon is in the quadratures, the sun's influence upon the waters where he is verti- cal, is an equipoise to that, where he is at the nadir. In these estimates of the measure of the sun's influence in the production of tides upon different parts of the earth's surface, and in various posi- tions of the moon in her orbit, it is not to be SOLAR INFLUENCE. 57 supposed that the co-efficients given can corre- spond with the height of the tides ; this will be modified by a cause which opposes a rise of the waters, and which increases in a more than simple ratio with the influences which produce tides, viz., the gravity of the waters. It being demonstrated that the moon, when near the syzygies, renders the sun's power to raise the waters upon opposite sides of the earth unequal, it follows from the earth's rotation, that two successive tides must at such times be un- equal. The declination of the sun and moon have an influence upon the height of two successive tides, and the laws which govern this inequality have been investigated by Newton and other philoso- phers, but they are admitted to be imperfectly understood. It is manifest that a known and an unknown cause, both operating to produce a tidal inequality, whose cycle is one day, might have been con- founded in their effects ; and it is possible that this demonstration may lead to a satisfactory explana- tion of some of the phenomena of the diurnal in- 58 THEORY OF THE TWO FORCES. equality, which, by exclusion, have been ascribed to local circumstances. This essay is not intended as a compendium of all that is knoVn upon the subject of the tides ; the aim of the writer has been directed to the following special points : Firstly. To show the insufficiency of the theory of simple attraction to explain the cause of the tide which occurs when the luminaries are at the nadir. Secondly. To point out the fact that there is in the different parts of the earth an unequal distri- bution of the centrifugal force, as well as of the attraction of the luminaries. Thirdly. From the inequality of the central forces, taken in connexion with the earth's rota- tion, to show the necessity of a periodical flux and reflux of the sea. Fourthly. To explain the effect which the moon's position in her orbit has upon the sun's power to raise the waters upon the opposite sides of the earth ; and to point to this last as a cause likely to perplex an observer of the diurnal ine- quality of the tides ; lastly, to suggest the proba- SOLAR INFLUENCE. 59 bility that this cause of inequality may, in con- nexion with that already known, viz., the decli- nation of the luminaries, give an explanation to some phenomena which have hitherto defied it. If the arguments used to sustain these views are deemed conclusive, the author is satisfied that his labor has not been wasted. APPENDIX, NOTES by the Mathematical Friend of the New England Editors, fyc. TEXT PAGE 16. " HERE then are two forces tending to oppose a rise of the waters at N." NOTE 1st. " Tending to oppose a recession of N from M ; but one of which favors a recession of O from N ; Provided, N and O are both free to move. A rise of water is therefore produced." Answer. The note involves .a provision that does not exist, viz., " That N and O are both free to move." The waters at N are moveable by reason of their fluidity, but O, the centre of the earth, is fixed in relation to its distance from the moon by the equilibrium of the central forces. See page 33. 62 APPENDIX. TEXT PAGE 17. "There can be no greater absurdity than the idea of the effect of a force being reversed, merely because its intensity is diminished." NOTE 2nd. "Is it not supposed 'reversed,' but being greater on O (and the solid mass) than on N, the distance ON is increased, although both N and O obey the force in its own direction." Answer. The theory which it is the purpose of this paper to combat, undertakes to show that the moon's attraction occasions two tides upon the earth's surface, one in the direction of her own mass, and the other away from it. It was in reference to this, that the expression, "reversed effect," has been used. TEXT PAGE 18. " Now, against this recession of the solid *mass of the earth, by which its waters are left behind it, it must be urged that the velocity of falling bodies upon the surface of the earth is a matter thoroughly understood ; consequently, the rate at APPENDIX. 63 which the earth must recede, in order tKat its waters should not overtake it, is an affair of calculation." NOTE 3rd. " A particle of the ocean is not a ' falling body,' except (as the whole earth is) in relation to the moon or sun. In relation to the earth, every such particle is in equilibrium till disturbed by some extraneous force. The solid mass is not obliged, if it recede, to move faster than a body at the surface would fall, if such is the idea in the manuscript." Answer. That a "particle of the ocean" is not " a falling body," in the ordinary acceptation of the term, is what every one knows ; and yet it is necessary to use the expression to point out the fallacy of Dr. Lardner's reasoning, when he says, " The solid mass of the earth D H, will, as it were, recede from the waters at N, in virtue of the excess of attraction, leaving those waters behind it, &c." Waters so " left behind," would fall in obedience to the laws of gravitation ; and further, as in the recession of the earth towards 64 APPENDIX. the moon by which its waters are left behind it, the waters would follow their parent orb with a velocity increasing as the square of the time, the mass of the earth must approach the moon at a greater velocity, in order to keep its waters behind it. To what absurdity does this theory lead us. The earth's nadir waters cannot be sustained by a negative cause ; there is a positive cause for the nadir tide ; but this the Newtonian theory ignores, for the whole edifice rests on the assertion that the moon revolves in an ellipsis, whose focus is in the centre of the earth. TEXT PAGE 20. "By this force alone, he (Sir Isaac Newton) says, the body P would describe around the body T, by the radius PT, areas proportional to the times, and an ellipsis whose focus is in the centre of the body T." NOTE 4th. "An ellipsis not in absolute space, but rela- tively to T supposed to be fixed, although, in APPENDIX. 65 fact, it "moves. " Circum se mutuo" is the ex- pressed idea. See XXIst Theorem. Answer. In the investigation of the causes of natural phenomena, it is generally unwise, and always dangerous to take as truth what is known to be false. The author of the marginal notes admits that "the body T in fact moves," and yet it is a part of Sir Isaac Newton's theory to sup- pose it fixed. TEXT PAGE 20. " Here is the error, for it is at variance with the laws of nature, for a body to describe an ellipsis, whose focus is in the centre of another body." NOTE 5th. "Is it an error then, to say that the earth describes (relatively) an ellipse, the sun being centrally in the focus?" See Principia, Propo- sition 57. Answer. The sun, the planets and their satel- lites, revolve about a point known as the common centre of the solar system ; and yet it is true that 5 66 APPENDIX. the earth and the other planets do revolve rela- tively in ellipses, the sun being centrally in one of the foci. It is equally true that the sun describes relatively an ellipsis about each of the planets, the latter being centrally in the focus. But do such assertions make out ideas of the motions of the heavenly bodies more clear ? Do they not rather tend to confuse them ? If so, why assert them ? But -most particularly, why build theories upon such assertions as if they were absolute truths ? TEXT PAGE 23. "This quotation, from the Principia, is made to prove that the views of its author, concerning the cause of the tides, were similar to those now gene- rally entertained ; and, also, more particularly to show that, in Corollaries 18 and 19 of Prop. LXVL, in which he applies the laws which govern the three bodies P, S and T, to the ex- planation of the flux and reflux of the sea, and in which the bodies T and S are intended to repre- sent the earth and moon, not the slightest allusion is made to the fact that they revolve about a centre common to the two." APPENDIX. 67 NOTE 6th. " Since the forces involved do not proceed from the centres of gravity, but from the centres of bodies, Newton, of course, reasoned from the figures described by the three bodies relatively." "Circum se mutuo." Answer. This explanation of Newton's mean- ing does not make his error (in founding his theory upon the idea, that the moon describes an ellipsis whose focus is in the centre of the earth) less fatal. TEXT PAGE 24. "It is remarkable, that Sir Isaac Newton, in his exposition of the tides, should not have availed himself of the ingenious truth suggested by Dr. Wallis ;* a truth which could not be demonstrated by the mind which conceived it for want of data, which the Principia afterwards supplied."! NOTE 7th. * " Not existing as a truth till brought out as an incident to Newton's own theory, and from which it could have been known independently, 68 APPENDIX. a true explanation could not have been deduced without Newton's discovered truths ; so that it would have changed, not the theory, but its form of presentation only." NOTE 8th. t " For want of principles and reasonings, which the author of the Principia brought to light." Answer. An -hypothesis may be true though not demonstrable, and such were the views of Dr. Wallis upon the subject of the tides. It is still remarkable that Sir Isaac Newton, with data at his command, possessed by himself alone, instead of showing the truth of Dr. Wallis' views, should have asserted conditions which are true only in a relative, sense, and attempted the construction of a theory thereon, which, from errors, inherent in its foundation, cannot bear investigation. NOTE 9th. ("From the preceding marginal notes, it will be concluded that this manuscript cannot differ in any true deductions or results from the ordinary APPENDIX. 69 theory, but can only, at best, offer them under some new process (if such it be) of demonstration or induction.") Answer. The last sentence invites the inference that the " mathematical friend" of the New Eng- land editors did not read the manuscript. If so, it was unwise to hazard an estimate of its value upon a basis so narrow as the one he has offered. An examination of these notes, and the replies to them, will make it apparent that Newton's error consisted in having made an assertion which, while relatively true, was absolutely at variance with fact ; and in having built a theory thereon, as if the conditions which he had asserted were actual truths. The result is, that the whole theory is defective. It may be convenient for some purposes to assume that the moon revolves about the centre of the earth, and that the latter does not share in the monthly revolution ; but a consequence of this would be that the earth could have no tangential force, and without a tangential force, there could be no nadir tide. In order to explain all the 70 APPENDIX. phenomena of the tides, it is necessary to describe the motions of the earth and moon with strict adherence to absolute truth. A criticism was made upon the manuscript by a gentleman occupying a high position in an institution of learning, in this city, which may be conveniently answered here. " The theory is ingenious, but I think I can reconcile it at all points with the Newtonian." One class of Newton's Expositors say the earth is "as it were falling towards the moon." Now, if so falling, there w r ould be a nadir tide from the diminished action of the moon upon the nadir waters, their inertia causing them to lag behind their parent earth. But, as facts do not justify this description of the earth's relation to the moon, any attempt to overthrow a theory founded upon it ought to be superfluous. Another class of Newton's followers describe the moon as revolving about the earth's centre as a focus, but abandon the fiction of the earth's falling towards the moon. These assert that the moon's diminished action upon the earth's nadir waters causes a rise. But this is impossible ; for APPENDIX. 71 her diminished action could have only a negative effect until some positive force is invoked to produce a rise of the waters. This positive force the essay shows to be the earth's centrifugal force, which cannot exist in the Newtonian theory, from the fact of the earth not sharing in the moon's monthly revolution. THE END. BBH i QO z^ t Co otfe? CN ,0 Z o tz CQ JJ = 00 ^. [X,,. ID \ Q Q o 5 ^c^