The reader will probably be making some comparison in his own mind of the deductions from this theory with the actual state of things. He will find some considerable resemblances; but he will also find such great differences as will make him very doubtful of its justness. In very few places does the high water happen within three-fourths of an hour of the moon's southing, as the theory leads him to expect; and in no place whatever does the spring tide fall on the day of new and full moon, nor the neap tide on the day of her quadrature. These always happen two or three days later. By comparing the difference of high water and the moon's southing in different places, he will hardly find any connecting principle. This shows evidently that the cause of this irregularity is local, and that the justness of the theory is not affected by it. By considering the phenomena in a navigable river, he will learn the real cause of the deviation. A flood tide arrives at the mouth of a river. The true theoretical tide differs in no respect from a wave. Suppose a spring tide actually formed on a fluid sphere, and the sun and moon then annihilated. The elevation must sink, pressing the under waters aside, and causing them to rise where they were depressed. The motion will not stop when the surface comes to a level; for the waters arrived at that position with a motion continually accelerated. They will therefore pass this position as a pendulum passes the perpendicular, and will rise as far on the other side, forming a high water where it was low water, and a low water where it was high water; and this would go on for ever, oscillating in a time which mathematicians can determine, if it were not for the viscidity, or something like friction, of the waters. If the sphere is not fluid to the centre, the motion of this wave will be different. The elevated waters cannot sink without diffusing themselves likewise, and occasioning a great horizontal motion in order to fill up the hollow at the place of low water. This motion will be greatest about half way between the places of high and low water. The shallower we suppose the ocean the greater must this horizontal motion be. The resistance of the bottom (though perfectly smooth and even) will greatly retard all the way to the surface. Still, however, it will move till all be level, and will even move a little farther and produce a small flood and ebb where the ebb and flood had been. Then a contrary motion will obtain; and, after a few oscillations, which can be calculated, it will be insensible. If the bottom of the ocean (which we still suppose to cover the whole earth) be uneven, with long extended valleys running in various directions, and with elevations reaching near the surface, it is evident that this must occasion great irregularities in the motion of the undermost waters,

both in respect of velocity and direction, and even occasion small inequalities on the surface, as we see in a river with a rugged bottom and rapid current. The deviations of the under currents will drag with them the contiguous incumbent waters, and thus occasion great superficial irregularities. Now a flood, arriving at the mouth of a river, must act precisely as this great wave does. It must be propagated up the river (or along it, even though perfectly level) in a certain time, and we shall have high water at all the different places in succession. This is distinctly seen in all rivers. It is high water at the mouth of the Thames at three o'clock, and later as we go up the river, till at London Bridge we have not high water till three o'clock in the morning, at which time it is again high water at the Nore. But in the mean time there has been low water at the Nore, and high water about half way to London; and, while the high water is proceeding to London, it is ebbing at this intermediate place, and is low water there when it is high water at London and at the Nore. Did the tide extend as far beyond London as London is from the Nore, we should have three high waters with two low waters interposed. The most remarkable instance of this kind is the Maragnon or Amazon river in South America. It appears, by the observations of Condamine and others, that between Para at the mouth of the river, and the conflux of the Madera and Maragnon, there are seven co-existent high waters, with six low waters between them. Nothing can more evidently show that the tides in these places are nothing but the propagation of a wave. The velocity of its superficial motion, and the distance to which it will insensibly go, must depend on many circumstances. A deep channel and gentle acclivity will allow it to proceed much farther up the river, and the distance between the successive summits will be greater than when the channel is shallow and steep. If we apply the ingenious theory of Chevalier Buat, we may tell both the velocity of the motion and the interval of the successive high waters. It may be imitated in artificial canals, and experiments of this kind would be very instructive. We have said enough at present for our purpose of explaining the irregularity of the times of high water in different places, with respect to the moon's southing. But we now see clearly, that something of the same kind must happen in all great arms of the sea which are of an oblong shape, and communicate by one end with the open ocean. The general tide in this ocean must proceed along this channel, and the high water will happen on its shores in succession. This also is distinctly seen. The tide in the Atlantic Ocean produces high water at new and full moon, at a later and later hour along the south coast of Great Britain in proportion as we proceed from Scilly Islands to Dover. In the same manner it is later and later as we come along the east coast from Orkney to Dover. Yet even in this progress there are considerable irregularities owing to the sinuosities of the shores, deep indented bays, prominent capes, and extensive ridges and valleys iu the channel. A similar progress is observed

along the coasts of Spain and France, the tide advancing gradually from the south, turning round cape Finnisterre, running along the north coast of Spain, and along the west and north coasts of France. The attentive consideration of these facts will not only satisfy us with respect to this difficulty, but will enable us to trace a principle of connexion amidst all the irregularities that we observe. And if we note the difference between the time of high water of spring tide as given by theory for any place, and the observed time of high water, we shall find this interval to be very nearly constant through the whole series of tides during a lunation. Suppose this interval to be forty hours. We shall find every other phenomenon succeed after the same interval. And, if we suppose the moon to be in the place where she was forty hours before, the observation will agree pretty well with the theory, as to the succession of tides, the length of tide day, the retardations of the tides, and their gradual diminution from spring to neap tide. We say pretty well; for there still remain several small irregularities, different in different places, and not following any observable law. These are therefore local, and owing to local causes, as we shall point out. There is also a general deviation of the theory from the real series of tides. The neap tides, and those adjoining, happen a little earlier than the corrected theory points out. Thus at Brest (where more numerous and accurate observations have been made than at any other place in Europe), when the moon changes precisely at noon, it is high water at 3 h. 38. When the moon enters her second quarter, at noon, it is high water at 8h. 40′, instead of 9h. 48′, which theory assigns. Something similar, and within a very few minutes equal to this, is observed in every place on the sea-coast. This is therefore something general, and indicates a real defect in the theory. But this arises from the same cause with the other general deviation, viz. that the greatest and least tides do not happen on the days of full and half moon, but a certain time after. We shall attempt to explain this. We set out with the supposition that the water acquired in an instant the elevation competent to its equilibrium. But this is not true. No motion is instantaneous, however great the force; and every motion, and change of motion, produced by a sensible or finite force, increases from nothing to a sensible quantity by infinitely small degrees. Time elapses before the body can acquire any sensible velocity; and, in order to acquire the same sensible velocity by the action of different forces acting similarly, a time must elapse inversely proportional to the force. An infinitely small force requires a finite time for communicating even an infinitely small velocity; and a finite force, in an infinitely small time, communicates only an infinitely small velocity; and, if there be any kind of motion which changes by insensible degrees, it requires a finite force to prevent this change. Thus a bucket of water, hanging by a cord lapped round a light and easily moveable cylinder, will run down with a motion uniformly accelerated; but this motion will be prevented by hanging an equal bucket on the other side, so as to act with a

finite force. This force prevents only infinitely small accelerations. It is therefore an infinitesimal of the first order, and may be restored in an instant, or the continuation of the depression prevented by a certain finite force. The weight of the rope makes it hang in an oblong curve, just as the force of the moon raises the waters of the ocean. Turn the rollers slowly, and the rope unwinding at one side, and winding up on the other side of the roller, will continue to form the same curve; but turn the roller the other way very briskly, and the rope will now hang in a curve considerably from the perpendicular; so that the force of gravity may in an instant undo the infinitely small elevation produced by the turning. This phenomenon has puzzled many persons not unaccustomed to such discussions. But another view of this matter leads to the same conclusion. There can be no doubt that the interval between high and low water is not sufficient for producing all the accumulation necessary for equilibrium in an ocean so very shallow. The horizontal motion necessary for gathering together so much water along a shallow sea would be prodigious. Therefore it never attains its full height; and when the waters, already raised to a certain degree, have passed the situation immediately under the moon, they are still under the action of accumulating forces, although these forces are now diminished. They will continue rising, till they have so far passed the moon that their situation subjects them to depressing forces. If they have acquired this situation with an accelerated motion, they will rise still farther by their inherent motion, till the depressing forces have destroyed all their acceleration, and then they will begin to sink again. It is in this way that the nutation of the earth's axis produces the greatest inclination, not when the inclining forces are greatest, but three months after. It is thus that the warmest time of the day is a considerable while after noon, and that the warmest season is considerably after midsummer. The warmth increases till the momentary waste of heat exceeds the momentary supply. Hence we conclude that it may be demonstrated that, in a sphere fluid to the centre, the time of high water cannot be less, and may be more, than three lunar hours after the moon's southing. As the depth of the ocean diminishes, this interval also diminishes. It is perhaps impossible to assign the distance at which the summit of the ocean may be kept while the earth turns round its axis. We can only see that it must be less when the accumulating force is greater, and therefore less in spring tides than in neap tides; but the difference may be insensible. All this depends on circumstances with which we are little acquainted; many of these circumstances are local: and the situation of the summit of the ocean, with respect to the moon, may be different in different places. Nor have we been able to determine theoretically what will be the height of the summit. It will certainly be less than the height necessary for perfect equilibrium. The result of all this reasoning is, that we must always suppose the summit of the tide is at a certain distance east of the place assigned by the theory. Mr. Bernoulli

concludes, from a very copious comparison of observations at different places, that the place of bigh water is about 20° to the eastward of the place assigned by the theory. Therefore the table formerly given will correspond with observation, if the leading column of the moon's elongation from the sun be altered accordingly. We insert it again in this place, with this alteration, and add three columns for the times of high water. Thus changed it will be of great use. We have now an explanation of the acceleration of the neap tides, which should happen six hours later than the spring tides. They are in fact tides corresponding to positions of the moon, which are 20° more, and not the real spring and neap tides. These do not happen till two days after; and, if the really greatest and least tides be observed, the least will be found six hours later than the

This table is general; and exhibits the times of high water, and their difference from those of the moon's southing, in the open sea, free from all local obstructions. If therefore the time of high water in any place on the earth's equator (for we have hitherto considered no other) be different from this table (supposed correct), we must attribute the difference to the distinguishing circumstances of the situation. Thus every place on the equator should have high water on the day that the moon, situated at her mean distance, changes precisely at noon, at 22′ P. M., because the moon passes the meridian along with the sun by supposition. Therefore, to make use of this

table, we must take the difference between the first number of the column, entitled time of high water, from the time of high water at full and change peculiar to any place, and add this to all the other numbers of that column. This adapts the table to the given place. Thus, to know the time of high water at Leith when the moon is 50° east of the sun, at her mean distance from the earth, take 22′ from 4 h. 30', there remains 4:08. Add this to 2 h. 48', and we have 6 h. 56' for the hour of high water. (The hour of high water at new and full moon for Edinburgh is marked 4 h. 30' in Maskelyne's tables, but we do not pretend to give it as the exact determination. This would require a series of accurate observations). It is by no means an easy matter to ascertain the time of high water with precision. It changes so very slowly, that we may easily mistake the exact minute. The best method is to have a pipe with a small hole near its bottom, and a float with a long graduated rod. The water gets in by the small hole, and raises the float, and the smallness of the hole prevents the sudden and irregular starts which waves would occasion. Instead of observing the moment of high water, observe the height of the rod about half an hour before, and wait after high water till the rod comes again to that height: take the middle between them. The water rises sensibly half an hour before the top of the tide, and quickly changes the height of the rod, so that we cannot make a great mistake in the time. Bernoulli has made a very careful comparison of the theory thus corrected, with the great collection of observations preserved in the Depôt de la Marin. at Brest and Rochefort (see Cassini Mem. Acad. Paris, 1734); and finds the coincidence very great, and far exceeded any rule which he had ever seen. Indeed we have no rules but what are purely empirical, or which suppose a uniform progression of the tides. The heights of the tides are much more affected by local circumstances than the regular series of their times. The regular spring tide should be to the neap tide in the same proportion in all places; but nothing is more different than this proportion. In some places the spring tide is not double the neap tide, and in other places it is more than quadruple. This prevented Bernoulli from attempting to fix the proportion by means of the height of the tides. Newton had, however, done it by the tides at Bristol, and made the lunar force almost five times greater than the solar force. But this was very ill-founded, for the reason now given. Yet Bernoulli saw that in all places the tides gradually decreased by a similar law with the theoretical tides, and has given a very ingenious method of accommodating the theory to any tides which may be observed. The result cannot be far from the truth. The following table is calculated for the three chief distances of the moon from the earth :

This table is corrected from the retardation arising from the inertia of the waters. Thus, when the moon is 20° from the sun, the mean distance tide is 1·00 A + 0·00 B, which is the theoretical tide corresponding to conjunction or opposition.

We have now given in sufficient detail the phenomena of the tides along the equator, when the sun and moon are both in the equator, showing both their times and their magnitude. When we recollect that all the sections of an oblong spheroid by a plane passing through an equatorial diameter are ellipses, and that the compound tide is a combination of two such spheroids, we perceive that every section of it through the centre, and perpendicular to the plane in which the sun and moon are situated, is also an ellipse, whose shorter axis is the equatorial diameter of a spring tide. This is the greatest depression in all situations of the luminaries; and the points of greatest depression are the lower poles of every compound tide. When the luminaries are in the equator, these lower poles coincide with the poles of the earth. The equator, therefore, of every compound tide is also an ellipse, the whole circumference of which is lower than any other section of this tide, and gives the place of low water in every part of the earth. In like manner, the section through the four poles, upper and lower, gives the place of high water. These two sections are terrestrial meridians or hour circles, when the luminaries are in the equator. Hence it follows that all that we have already said as to the times of high and low water may be applied to every place on the surface of the earth, when the sun and moon are in the equator. But the heights of tide will diminish as we recede from the equator. The heights must be reduced in the proportion of radius to the cosine of the latitude of the place. But in every other situation of the sun and moon all the circumstances vary exceedingly. It is true that the determination of the elevation of

the waters in any place whatever is equally easy. The difficulty is to exhibit for that place a connected view of the whole tide, with the hours of flood and ebb, and the difference between high and low water. This is not indeed difficult; but the process by the ordinary rules of spherical trigonometry is tedious. When the sun and moon are not near conjunction or opposition, the shape of the ocean resembles a turnip, which is flat and not round in its broadest part. Before we can determine with precision the different phenomena in connexion, we must ascertain the position or attitude of this turnip; making on the surface of the earth both its elliptical equators. One of these is the plane passing through the sun and moon, and the other is perpendicular to it, and marks the place of low water. And we must mark in like manner its first meridian, which passes through all the four poles, and marks on the surface of the earth the place of high water. The position of the greatest section of this compound spheroid is often much inclined to the earth's equator; nay, sometimes is at right angles to it when the moon has the same right ascension with the sun, but a different declination. In these cases the ebb tide on the equator is the greatest possible; for the lower poles of the compound spheroid are in the equator. Such situations occasion a very complicated calculus. We must therefore content ourselves with a good approximation. And first, with respect to the times of high water. It will be sufficient to conceive the sun and moon as always in one plane, viz. the ecliptic. The orbits of the sun and moon are never more inclined than 5° 30′. This will make very little difference; for, when the luminaries are so situated that the great circle through them is much inclined to the equator, they are then very near to each other, and the form of the spheroid is little different from what it would be if they were really in conjunction or opposition. It will

therefore be sufficient to consider the moon in three different situations. 1. In the equator. The point of highest water is never farther from the moon than 15°, when she is in apogee and the sun in perigee. Therefore, to have the time of high water, multiply the numbers of the columns which express the difference of high water and the moon's southing by, and the products give the real difference. It is more difficult to find the time of low water; and we must either go through the whole trigonometrical process or content ourselves with a less perfect approximation. The trigonometrical process is not indeed difficult. But it will be abundantly exact to consider the tide as accompanying the moon only. Hence the two tides of one lunar day may be considerably different, and it is proper to distinguish them by different names, as a superior tide which happens when the moon is above the horizon during high water, and the inferior tide. From this construction we may learn in general, 1. When the moon has no declination, the durations and also the heights of the superior and inferior tides are equal in all parts of the world. 2. When the moon has declination, the duration and also the height of a superior tide at any place is greater than that of the inferior; or is less than it, according as the moon's declination and the latitude of the place are of the same or opposite names. This is a very important circumstance. It frequently happens that the inferior tide is found the greatest when it should be the least; which is particularly the case at the Nore. This shows, without further reasoning, that the tide at the Nore is only a branch of the regular tide. The regular tide comes in between Scotland and the continent; and, after travelling along the coast, reaches the Thames, while the regular tide is just coming in again between Scotland and the continent. 3. If the moon's declination is equal to the co-latitude of the place, or exceeds it, there will be only one tide in a lunar day. It will be a superior or inferior tide, according as the declination of the moon and the latitude of the place are of the same or opposite kinds. Thus the difference of the durations of the superior and inferior tides of the same day increase both with the moon's declination and with the latitude of the place. The heights of the tides are affected no less remarkably by the different situations of the moon, and of the place of observation. Therefore at high water, which by the theory is in the place directly under the moon, the height of the tide is as the square of the cosine of the moon's zenith or nadir distance. Hence we derive a construction which solves all questions relating to the height of the tides with great facility, free from all the intricacy and ambiguities of the algebraic analysis employed by Bernoulli. 1. The greatest tides happen when the moon is in the zenith or nadir of the place of observation. 2. When the moon is in the equator, the superior and inferior tides have equal height. 3. If he place of observation is in the equator, the inferior and superior tides are again equal, whatever is the moon's declination. 4. The superior tides are greater or less than the inferior tides according as the latitude and declination are of

the same or of opposite names. 5. If the co-latitude be equal to the declination, or less than it, there will be no inferior tide, or no superior tide, according as the latitude of the place, and declination of the moon, are of the same or opposite names; and the low water of its only tide is the summit of the inferior tide. 6. At the pole there is no daily tide; but there are two monthly tides, and it is low water when the moon is in the equator. N. B. The moon's declination never exceeds 30°. Therefore cos. 2 M Q is always a positive quantity, and never less than half, which is the cosine of 60°. While the latitude is less than 45°, cos. 2 lat. is also a positive quantity. When it is precisely 45° the cosine of its double is o; and, when it is greater than 45°, the cosine of its double is negative. Hence we see, 1. That the medium tides are equally affected by the northern and southern declinations of the moon. 2. If the latitude of the place is 45°, the medium tide is always M. This is the reason why the tides along the coasts of France and Spain are so little affected by the declination of the moon. 3. If the latitude is less than 45°, the mean tides increase as the moon's declination diminishes. All that we have now said may be said of the solar tide. Also the same things hold true of spring tides. But to ascertain the effects of declination and latitude on other tides, we must make a much more complicated construction, even though we suppose both luminaries in the ecliptic. For in this case the two depressed poles of the watery spheroid are not in the poles of the earth; and therefore the sections of the ocean, made by meridians, are by no means ellipses. The inaccuracies are not so great in intermediate tides, and respect chiefly the time of high water and the height of low water. The exact computation is very tedious and peculiar, so that it is hardly possible to give any account of a regular progress of phenomena; and all we can do is, to ascertain the precise heights of detached points. For which reasons we must content ourselves with the construction already given. It is the exact geometrical expression of Bernoulli's analysis, and its consequences now related contain all that he has investigated. Thus have we obtained a general, though not very accurate, view of the phenomena which must take place in different latitudes and in different declinations of the sun and moon, provided that the physical theory which determines the form and position of the watery spheroid be just. We have only to compute, by a very simple process of spherical trigonometry, the place of the pole of this spheroid. If we were to compare this theory with observation, without further consideration, we should still find it unfavorable, partly in respect of the heights of the tides, and more remarkably in respect of the time of low water. We must again consider the effects of the inertia of the waters, and recollect that a regular theoretical tide differs very little in its progress from the motion of a wave. Even along the free ocean, its motion much resembles that of any other wave. All waves are propagated by an oscillatory motion of the waters, precisely similar to that of a pendulum. It is well known that if a pendulum